Minisymposia
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MS01 Recent Progress in Partial Differential and Difference Equations Main organizer: Takiko Sasaki (Musashino University) · 5 talks
Organizers
Main organizer: Takiko Sasaki (Musashino University)
- Co-organizer: Tetsuji Tokihiro (Musashino University)
Program & Speakers
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On lifespan estimates for a discrete Fujita equation
Kohei Higashi (Musashino University)
We study the lifespan of solutions to a discrete analogue of the semilinear heat equation with power nonlinearity, viewed as a discrete Fujita equation on the lattice. Previous work has shown that this model exhibits finite-time blow-up with the same Fujita critical exponent as in the continuous setting. We investigate how the lifespan depends on the size of small initial data and on the exponent relative to the Fujita critical exponent. We prove that, in both the subcritical and critical cases, the lifespan has the same order as that of the corresponding continuous problem.
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Global solution structure for a cell polarization model with a finite diffusion coefficient
Tatsuki Mori (Musashino University)
We investigate the global bifurcation structure of solutions for a nonlinear boundary value problem with a nonlocal constraint arising in a cell polarization model with mass conservation proposed by Mori, Jilkine, and Edelstein-Keshet. Regarding the shadow system of the stationary problem, a representation formula for the solutions and the global bifurcation diagrams have already been obtained. In this talk, we derive a representation formula for the solutions of the original problem and report on numerical simulations of its global bifurcation diagram. In particular, global bifurcation diagrams include secondary and imperfect bifurcation phenomena.
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Diffuse Adiabatic Flows in Thermally Coupled Grounded Shallow Ice Sheets: Modelling and Analysis
Paolo Piersanti (The Chinese University of Hong Kong, Shenzhen)
In this talk we propose a novel thermodynamical model which couples the evolution of the thickness of a grounded shallow ice sheet with the evolution of its internal temperature. Both the grounded shallow ice sheet surface elevation and the ice internal temperature are subjected to physical constraints. The equations governing the evolution of the grounded shallow ice sheet thickness are degenerate, and the ice internal temperature evolves in a moving domain. First, we formally model the phenomenon under consideration by adopting strategies akin to those employed in the construction of diffuse-interface models. Second, we establish the existence of solutions for one such formal model by means of the penalty method, and we observe that the low regularity of the problem under consideration prevents us from obtaining a standard concept of solution.
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A Structure-Preserving Stagewise Rescaling Algorithm for a Two-Dimensional Nonlocal MEMS Equation
TETSUJI TOKIHIRO (Musashino University)
Nonlocal MEMS equations can exhibit finite-time quenching, or touchdown, which is challenging to resolve numerically. We study a stagewise rescaling algorithm for a two-dimensional nonlocal MEMS equation in an asymptotically constant-feedback touchdown regime. Although the equation is not exactly invariant under the $(A^{3/2})–(A^3)$ scaling used here, the scaling is justified when the reciprocal-integral feedback remains bounded and converges to a positive finite limit, reducing the leading core dynamics to a local MEMS equation with an asymptotically constant coefficient. We prove exact energy dissipation within each fixed stage, derive a discrete energy inequality using a minimizing-movement solver, and quantify switch and outer-update defects. This yields almost monotonicity and a defect-aware criterion for nonexistence of global admissible continuation. Numerical tests illustrate the method and identify diagnostics needed for a posteriori verification.
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Single-Point Touchdown for Radially Symmetric Solutions of a Supercritical Nonlocal MEMS Equation
Takiko Sasaki (Musashino Unversity)
We study finite-time touchdown for a radially symmetric nonlocal MEMS equation in the supercritical regime. Our aim is to construct solutions that touch down only at the origin and whose final profile has a prescribed higher-order algebraic form. The key change of variables is to use $F=(W+\eta)^{-(p+1)}$ together with a contracting radial coordinate $z$; in these variables the leading profile becomes $F_B(z)=\alpha_m^{-1}+Bz^m$. This formulation separates the lower-order Taylor modes, the modulation parameter $B$, and the stable infinite-dimensional remainder. The nonlocal term is controlled by incorporating its leading contributions into a finite-dimensional modulation system for the lower modes and by estimating the remaining stable part. Combining tail estimates, stable remainder estimates, outer barriers, and a degree argument, we obtain smooth radial initial data and the asymptotic profile $(1-v)^{p+1}\sim \Lambda_m\{\alpha_m^{-1}(T-t)^{A_m}+B_\infty |x|^m\}$.
MS02 Recent Topics in Applied Mathematics and Computation Main organizer: Peter Chang-Yi Weng (National Chiayi University) · 8 talks
Organizers
Main organizer: Peter Chang-Yi Weng (National Chiayi University)
Program & Speakers
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The refining estimates of invariant subspaces through projected nonsymmetric algebraic Riccati equations in high dimensionality reduction and image compression
Peter Chang-Yi Weng (National Chiayi University)
In this talk, we provide an efficient and accurate method, which is the numerical solution of the projected nonsymmetric algebaric Riccati equations (pNAREs) arisen in the refinement of estimates of invariant subspaces (REIS). The theoretical contribution of our paper is threefold. Firstly, we introduce how to apply SVD to do image compression. Secondly, the core of this paper is to adapt REIS and find the first singular values to do image compression. Thirdly, we describe some measurement tools like compression ratio, mean square error, peak signal to noise ratio and structural similarity index to evaluate the performance of the compression factors and the quality of the compressed images. Furthermore, the operation counts and computational complexity are also provided. Finally, we present some numerical experiments about some real world image dataset to show the feasibility of our proposed algorithm.
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Geometric spectral separation and algebraic dichotomy for critical Stein matrix equations
Chun-Yueh Chiang (National Formosa University)
This talk considers the critical Stein matrix equation $X-AXB=C$ with $\rho(A)\rho(B)=1$, which causes the $r$-Smith iteration to fail. We propose two structure-preserving preconditioning strategies to resolve this challenge. First, a geometric approach utilizes Moebius transformations to map the problem into a contractive domain. Under spectral separability, convergence is related to Apollonius circles. For interleaved eigenvalues on the unit circle, a power iteration strategy reorganizes the spectra into disjoint arcs to restore separability. Second, for pathological configurations, we develop a three-region algebraic dichotomy. The system is converted into a Sylvester equation via Cayley transform. By integrating shifted matrix sign functions with optimal even polynomials, we partition the spectra into orthogonal regions and shift imaginary eigenvalues into stable half-planes without subspace extraction.
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Quantum Channel Reconstruction
Matthew Lin (National Cheng Kung University)
In this work, we propose a manifold-constrained optimization framework for quantum channel reconstruction using only a single input–output pair. By employing projected gradient dynamics on the Stiefel manifold and the probability simplex, we recover the underlying channel parameters while preserving the intrinsic geometric constraints of the problem. Convergence of the proposed method is established. Numerical experiments demonstrate the effectiveness and flexibility of the approach, including its extension to multiple input–output pairs.
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A Characterization of Positive Entropy of Markov Tree-Shifts
Nai Zhu Huang (National University of Kaohsiung)
Topological entropy is often considered as an indicator of complexity. However, there is no finitely checkable algorithm to determine the positive entropy of multidimensional shifts of finite type. Indeed, it is right recursively enumerable. In this presentation, we will present the characterization of positive entropy in Markov tree-shifts by adjacency matrices.
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The Kelmans Transformation and the Largest Real Eigenvalue of Nonnegative Matrices
Louis Kao (Fu Jen Catholic University)
The Kelmans transformation is originally defined on graphs and their adjacency matrices, which has some nice properties on the spectrum of graphs. For a nonnegative square matrix $C$ of order $n$ and fix a $2$-subset $\{a, b\}\subseteq [n]$ with certain constrains, we define $C'(a,b)$ to be a general version of the Kelmans transformation of $C$. In this talk, we will show that the largest real eigenvalue of $C$ is at most the largest real eigenvalue of $C'(a,b)$. Furthermore, we will apply this tool to discuss the Hamiltonicity of mixed graphs, whose adjacency matrix is not necessary symmety.
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Nonlinear Modeling of Fishery Resources via Kolmogorov-Arnold Networks and Latent Variable Frameworks
Hung-Hsun Chen (Fu Jen Catholic University)
Reversing biometric relationships to predict shark length from weight is crucial for fisheries management, as market data often lacks size records. Standard neural networks (ANNs) capture these nonlinearities but act as black boxes. We propose a mathematically rigorous framework integrating Kolmogorov-Arnold Networks (KANs) with a latent variable formulation to reconstruct length from weight. We model ten Northwest Pacific shark species independently. Since practical market data lacks sex classification, sex is excluded as an input. To capture unobserved physiological dynamics like sexual dimorphism and age, a latent layer models these hidden covariates within the KAN. KANs employ learnable splines on edges, dramatically improving parameter efficiency and interpretability. Our KAN-latent framework matches ANN accuracy while enabling symbolic equation discovery for sustainable ecological management.
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State Transformation of Stock Prices and Its Applications
CHIEN-CHANG YEN (Mathematics, Fu Jen Catholic University)
A state on the time zone $[t_0,t_1]$ is defined by two successive values, $S_0=S(t_0)$ and $S_1=S(t_1)$ and the imposed equillibrium $A$. A state transform is to represent a discrete stock time series by determining the state at each time zone. We simulate four simple/fundamental states of a time series, increasing, decreasing, white noise, and random walk of time series, for revealing the relation between time series and states. The analysis of state transform for the aforementioned time series are shown in theoretical approach. It is significant that the behavior of white noise and random walk can be distinct by the state transform. The application on predicting state is used the weighted probability sample, which is also used in the two-gram language model, and the simulations show that the the nearest future states could be reasonablely acceptable through our simulations of financial markets data.
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Numerical Methods and Closed-Form Solutions for Structured Sylvester Equations in Pseudospectral Discretizations
Yung-Ta Li (Fu Jen Catholic University)
In this talk we first review the Bartel-Stewart algorithm that numerically solve the Sylvester equation and a closed-form finite series representation of the unique solution. Then we derive a structured Sylvester equation by discretizing a two dimensional advection equation using a pseudospectral mathod with penalty. To solve the structured Sylvester equation, we propose a numerical method that takes advantage of the skew-symmetric structure and we aim to seek for a closed-form solution to the structured Sylvester equation with particular right hand sides.
MS03 Numerical analysis and applications of nonlocal models Main organizer: Xiaobo Yin (Central China Normal University) · 9 talks
Organizers
Main organizer: Xiaobo Yin (Central China Normal University)
- Co-organizer: Jiwei Zhang (Wuhan University)
Program & Speakers
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A unified superconvergent postprocessing technique of the Galerkin methods for solving Volterra integro-differential equations
Lijun Yi (Shanghai Normal University)
In this talk I shall introduce a unified postprocessing technique of the continuous and discontinuous Galerkin methods applied to solve first and second-order Volterra integro-differential equations. The core idea of this postprocessing technique is to augment the existing Galerkin approximation of degree $k$ with an additional term involving a generalized Jacobi polynomial (multiplied by appropriate coefficients) of degree $k+1$. The theoretical findings suggest that, through postprocessing, the convergence rate of the original Galerkin approximation error can be elevated by one order. Abundant numerical results underscore the effectiveness and high accuracy of the proposed postprocessing technique.
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Error estimates of FEMs for nonlocal problems using exact or approximated interaction neighborhoods
Xiaobo Yin (Central China Normal University)
In this report, we study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.
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The high accuracy algorithms for finding multiple solutions to nonlinear partial differential systems
Zhaoxiang Li (Shanghai Normal University)
In this talk, we investigate the computation of multiple solutions to nonlinear partial differential systems, particularly in settings where global variational structures are unavailable. To overcome the limitations of classical methods, we develop a series of computational frameworks based on augmented singular transformations, partial Newton correction techniques and spectral discretizations. For semi-linear systems, we introduce an augmented singular transform and partial Newton correction method for the stable computation of multiple solutions. For multi-component systems with heterogeneous nonlinearities, we develop a modified partial Newton correction method based on a two-parameter scaling strategy to efficiently compute solutions with complex spatial structures. The framework is further extended to quasi-linear systems involving degenerate p-Laplacian-type operators, yielding a generalized partial Newton correction method that effectively addresses operator singularities.
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Complete Invariants of Symmetric and Self-Adjoint Matrix Families
Huajie Chen (Beijing Normal University)
A parameterized matrix family on a configuration domain is determined by its physical content only up to a constant orthogonal change of basis. This gauge ambiguity is intrinsic to data-driven Hamiltonian models, such as tight-binding parameterizations, reduced-order electronic structure methods, or excited states models. It raises a basic inverse problem: what observations suffice to identify the family up to this gauge? The pointwise spectrum is incomplete already for linear families. Here, we prove that, under natural non-degeneracy and connectivity assumptions, augmenting the spectrum with loop products of the coupling matrices in the instantaneous eigenframe yields a complete invariant. We support the theory with numerical experiments.
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Re-derivation and Mathematical Analysis for Linear Peridynamics Model for Arbitrary Poisson Ratio’s Material
Yufeng Nie (Northwestern Polytechnical University)
This talk is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson’s ratio material. Based on the fundamental laws of dynamics, we re-derive the bond-based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well-posedness of time-dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.
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Entropy-consistent numerical methods and singular limit for nonlocal conservation laws
Kuang Huang (The University of Hong Kong)
We consider a class of nonlocal conservation laws modeling traffic flow, where the velocity depends on a convolution of the density with a rescaled kernel that concentrates as a parameter $\epsilon$ tends to zero. A central question is whether solutions converge to the entropy-admissible solution of the corresponding local conservation law, and how to design numerical methods that faithfully capture this transition. We present numerical schemes that are asymptotically compatible: as the mesh size $h$ and $\epsilon$ vanish simultaneously, numerical solutions converge to the local entropy solution with an explicit rate of order $\sqrt{\epsilon}+\sqrt{h}$. We also discuss a complementary approach via compensated compactness that resolves the singular limit for rough initial data and kernels.
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A weighted quantile filter based framework for interface optimal design problems
Dong Wang (The Chinese University of Hong Kong, Shenzhen)
We present a robust and efficient numerical framework based on a median filter scheme for solving a broad class of interface optimal design problems, from image segmentation to topology optimization. A key innovation of our work is the extension of the binary scheme into a level-set scheme via a weighted quantile interpretation. Unlike traditional binary iterative convolution-thresholding method (ICTM), this continuous weighted quantile filter scheme effectively overcomes the pinning effect caused by spatial discretization, achieving interface evolution even with small time steps. We also provide a rigorous theoretical analysis, proving the unconditional energy stability of the iterative scheme. Furthermore, we prove that for a wide class of data fidelity terms, the convex relaxation inherently enforces a binary solution, justifying the effectiveness of the method without explicit penalization.
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Two-material optimal design problem governed by the heat equation
Tomoyuki Oka (Fukuoka Institute of Technology)
In this talk, we shall consider the two-material optimal design problem for the time-averaged duality pairing between a heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In the elliptic (i.e., time-independent) case, this problem can be relaxed through homogenization, and the relaxation problem has a self-adjoint structure. In this elliptic setting, a direct relaxation with respect to the density is possible, and the convexity of the relaxed problem allows us to construct an optimal density. On the other hand, in the parabolic case, such a structure is in general not ensured due to the time-dependence. In this talk, we shall discuss the direct relaxability and the long-time behavior for the parabolic case in some specific settings.
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Optimal design of two-material thermal conductors with thermal radiation
Kei Matsushima (Hiroshima University)
Thermal radiation is important in high-temperature and thermally isolated systems, but it leads to nonlinear boundary conditions beyond standard linear theories for optimal design. This talk presents an optimal design framework for two-material thermal conductors governed by steady-state diffusion equations with nonlinear boundary conditions, including radiation modeled by maximal monotone operators. We discuss a homogenization-based relaxation of the original topology optimization problem and show that the relaxed problem admits minimizers with the same infimum value as the original one. We then introduce a level-set approximation to avoid grayscale material distributions and derive sensitivities for numerical optimization. Numerical examples for radiative heat-dissipating structures illustrate how nonlinear boundary effects and heat-source intensity shape optimal configurations, and show that optimized designs can outperform intuitive fin-like structures.
MS04 Dispersive PDEs: Modeling, Analysis and Numerical Simulations Main organizer: Yue Feng (Xi'an Jiaotong University) · 5 talks
Organizers
Main organizer: Yue Feng (Xi'an Jiaotong University)
- Co-organizer: Weizhu Bao (National University of Singapore)
Program & Speakers
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Multiscale methods and analysis for the Dirac equation in nonrelativistic regime
Weizhu Bao (National University of Singapore)
In this talk, I will review our recent works on multiscale methods and analysis for solving the highly oscillatory (nonlinear) Dirac equation including the nonrelativistic regime. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite. Rigorous error bounds are obtained for finite difference time domain methods, time splitting Fourier pseudospectral method and exponential wave integrator Fourier pseudospectral method, which depend explicitly on the mesh size, time step and the small parameter. Then based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral method for the Dirac equation and prove its error bound which uniformly accurate in term of the small dimensionless parameter. This is a joint work with Yongyong Cai, Yue Feng, Xiaowei Jia, Qinglin Tang and Jia Yin.
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On a class of low regularity solutions to the Camassa—Holm equation
Xiang-Ke Chang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
The Camassa—Holm equation is a nonlinear wave equation that has attracted much attention because of the existence of peaked solitons (peakons) and being a meaningful model for wave breaking. We describe a class of conservative low regularity solutions to the Camassa—Holm equation on the line by exploiting the moment problem and generalized indefinite strings to develop the inverse spectral method. In particular, we identify explicitly the solutions that are amenable to this approach, which include solutions made up of infinitely many peakons. As an application, our results are then used to investigate the long-time behavior of solutions. We also present numerics for these solutions with certain step-like and asymptotically eventually periodic initial data, etc.
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Hydrodynamics limit and numerical simulation for the Chern-Simons-Schrodinger equations
Jeongho Kim (Kyung Hee University)
In this talk, I present the analytical and numerical results on the Chern-Simons-Schrodinger (CSS) equations. In the first part of the talk, we rigorously prove the semi-classical limit of the CSS equations when the Planck constant converges to zero. The resulting equation is the pressure-less Euler equation. In the second part of the talk, we discuss the numerical methods for solving the CSS equations, including finite difference method and time-splitting method, and verify the analytical result via numerical experients.
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Four-Fermion Interaction Approximation of the Intermediate Vector Boson Model
Yoshio Tsutsumi (Kyoto University)
I will talk about the approximation of the Dirac-Proca equations to the cubic Dirac equation when the mass of the Proca field is large enough. This justifies the four-fermion interaction approximation of the intermediate vector boson model in a mathematically rigorous sense. The proof is based on the Strichartz estimate of the Proca sector with mass parameter. My talk is based on the paper published in Bull. Braz. Math. Soc., New Series, 56-3 (2025).
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Explicit Symmetric Low-Regularity Integrator for the Nonlinear Schrodinger Equation
Yue Feng (Xi'an Jiaotong University)
The numerical approximation of low-regularity solutions to the nonlinear Schrodinger equation (NLSE) is notoriously difficult and even more so if structure-preserving schemes are sought. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for NLSE. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.
MS05 Recent Advances in AI for Scientific Computing Main organizer: Jinsil Lee (Seoul National University) · 7 talks
Organizers
Main organizer: Jinsil Lee (Seoul National University)
- Co-organizer: Youngjoon Hong (Seoul National University)
- Co-organizer: Myeongsu Lee (Seoul National University)
Program & Speakers
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Finite Element Operator Networks: Mathematical Theory and Numerical Computation
Seungchan Ko (Inha University)
In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success across a wide range of fields. Operator learning has emerged as a powerful framework for parametric PDE problems: by learning the underlying solution operator, it enables rapid solution predictions as the input data varies. In this talk, I will propose a new operator-learning method based on the finite element method, which is highly flexible across diverse settings without relying on any precomputed training data. Grounded in my recent results, the goal of this talk is to provide a broad mathematical investigation of the proposed method. First, through numerical experiments across a variety of scenarios, I will demonstrate practical effectiveness and versatility of the method. Second, leveraging mathematical theory for neural networks, I will develop a theoretical analysis of the approach and design theory-guided strategies that can substantially improve performance.
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Scientific Machine Learning: Methods and Applications
Wei-Fan Hu (National Central University)
In recent years, the advancement of machine learning techniques has achieved remarkable success and significantly influenced various scientific fields. It also provides a novel avenue for creating a new category of numerical methods to address solutions for partial differential equations that are either intractable or impractical using traditional numerical approaches. Despite these advancements, numerous computationally challenging problems remain unsolved within the scientific computing community. These challenges can potentially be addressed through the application of machine learning techniques. In this talk, we focus on developing numerical methods that integrate the strengths of classical numerical methods with modern deep learning approaches to tackle some of these problems of our primary interests.
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Transformers Subsume MLPs: Direct Transfer of Approximation Rates
Changhoon Song (Seoul National University)
Transformers have become a central architecture in modern machine learning, but their approximation theory has often been developed separately from the classical theory of multi-layer perceptrons. In this talk, I will present a simple emulation principle showing that Transformers can reproduce MLPs with only a constant-factor overhead in the number of parameters. As a consequence, any error rate established for MLPs transfers directly to Transformers without degradation in the rate. This provides a unified way to reinterpret several existing approximation results for Transformers and immediately yields new guarantees, including rates over Sobolev and Barron-type function classes. I will focus on the main idea of the construction--how attention can be used to aggregate token-wise information so as to emulate fully connected layers.
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Reduced-Order Modelling of Global Discretization Error Dynamics for High-Dimensional Linear ODEs via the Iterative Rational Krylov Method
ANNING LAI (The University of Osaka)
Local discretization errors of time-stepping schemes can often be estimated, whereas the corresponding global error is harder to obtain, especially for high-dimensional ODE systems and semi-discretized PDEs. We study reduced-order modelling of global discretization error dynamics under the assumption that local error information is available. The global error is formulated as a forced discrete-time dynamical system driven by the local error sequence.After POD compression of local errors and suitable reformulation, this model is rewritten as a discrete-time MIMO linear time-invariant (LTI)system.This allows the use of projection-based model reduction techniques developed for control systems. In particular, we adopt the Iterative Rational Krylov Algorithm to construct a low-dimensional approximation of the global error model.The talk focuses on linear ODE problems with stable error propagation, and briefly discusses an extension to nonlinear problems via parameter-dependent LTI systems.
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Point Cloud Neural Operator for Parametric PDEs on Complex and Variable Geometries
Daniel Huang (Peking University)
Surrogate models are critical for accelerating computationally expensive simulations in science and engineering, particularly for solving parametric partial differential equations (PDEs). Developing practical surrogate models poses significant challenges, particularly in handling geometrically complex and variable domains, which are often discretized as point clouds. In this work, we systematically investigate the formulation of neural operators---maps between infinite-dimensional function spaces---on point clouds to better handle complex and variable geometries while mitigating discretization effects. We introduce the Point Cloud Neural Operator (PCNO), designed to efficiently approximate solution maps of parametric PDEs on such domains. We evaluate the performance of PCNO on a range of pedagogical PDE problems, focusing on aspects such as boundary layers, adaptively meshed point clouds, and variable domains with topological variations.
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Galerkin Neural Networks-based Reduced Basis Method for Parametric Differential Equations
zhiping mao (Eastern Institute of Technology, Ningbo)
Reduced basis methods (RBM) are powerful tools for parametric PDEs, but their offline stage typically relies on high-fidelity solutions from mesh-dependent solvers, which can be expensive and non-flexible for sharp boundary problems. Recently, neural networks have emerged as an effective tool for approximating partial differential equations (PDEs). In particular, the residual-based Galerkin Neural Network (GNN) method adaptively generates basis functions to approximate variational equations, progressively improving accuracy. However, this adaptive approach requires retraining for each new parameter of the parametric PDEs. In this work, we propose a novel GNN-RBM method for parametric PDEs' approximation by establishing a theoretically rigorous connection between greedy parameter sampling and neural network-based adaptive basis generation. The resulting basis functions are valid across the entire parameter domain, including parameters not used during training.
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Physics-informed sampler for solver-free data generation in operator learning
Xuhui Meng (Huazhong University of Science and Technology)
Neural operator training requires large datasets from expensive numerical solvers, especially for nonlinear, high-dimensional, time-dependent, and geometrically varying PDEs. We propose a physics-informed sampler that bypasses solvers entirely: it defines a generative model over solution space using Bayesian neural network priors, samples candidate solutions, and derives source terms/BCs/ICs via automatic differentiation of governing operators. Evaluated on reaction-diffusion, Poisson (parameterized geometries), 2D/5D time-dependent PDEs, and vortex-induced vibration, the sampler generates diverse PDE-consistent training pairs at hundreds of times lower cost than conventional methods, supporting accurate neural operator predictions across broad test conditions.
MS06 Locally Conserved and Structure-Preserving Numerical Algorithms for Partial Differential Equations Main organizer: Shuyu Sun (Tongji University) · 8 talks
Organizers
Main organizer: Shuyu Sun (Tongji University)
- Co-organizer: Eun-Jae Park (Yonsei University)
Program & Speakers
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Stable auxiliary variable viscosity splitting method for the incompressible Navier-Stokes equations
Junxiang Yang (Macau University of Science and Technology)
In this talk, we develop an energy-stable, second-order accurate, and stable viscosity splitting method for the penalized incompressible Navier-Stokes equations. At each time step, it is sufficient to solve a Poisson equation to update the pressure and several linear elliptic equations with variable coefficients to update the velocity components. We analytically prove that the time-discretized energy satisfies the unconditional dissipation law. Moreover, we estimate that the L2-norm of velocity at each time step is bounded by the sum of the L2-norm of initial velocity and a constant, the H1-norm of velocity is weakly stable, i.e., it is bounded by a constant which is related to the time step. Extensive numerical experiments are implemented to verify the capability of our proposed method.
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A structure-preserving discretization for the coupled Stokes-Biot equations
Lina Zhao (City University of Hong Kong)
In this talk I will present a structure-preserving discretization for the coupled Stokes--Biot equations. The method is based on a fully-mixed formulation and computes directly the fluid and poroelastic stresses and the Darcy velocity. The scheme employs equal-order polynomial approximations for all unknowns, yet exhibits local momentum conservation, strongly symmetric stress fields, and exactly divergence-free velocity in the fluid domain. Transmission conditions enforcing mass conservation, stress balance, and the Beavers--Joseph--Saffman slip with friction law are incorporated directly into the formulation as natural interface conditions through careful balance of finite element spaces across subdomains. Several numerical experiments are carried out to validate the accuracy and robustness of the proposed scheme.
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Superconvergent and Divergence-Free Mixed Finite Element Methods for The Stokes Equation
Xuehai Huang (Shanghai University of Finance and Economics)
This paper develops superconvergent and divergence-free mixed finite element methods for the Stokes equation. We employ H(div)-conforming finite elements for the velocity and discontinuous piecewise polynomials for the pressure, ensuring that the divergence-free condition is satisfied pointwise. To discretize the vector Laplacian, we introduce a weak deviatoric gradient operator constructed from tangential--normal continuous traceless tensor elements. This approach yields a consistent and stable discretization without the need for additional stabilization terms required in discontinuous Galerkin or virtual element methods. We prove a discrete inf-sup condition and establish optimal-order error estimates for the stress. Notably, we derive superconvergent estimates for the velocity and pressure. The proposed framework provides a unified perspective on nonconforming virtual element methods and pseudostress-velocity-pressure formulations.
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An Adaptive Mixed CEM-GMsFEM for Two-Phase Flow in High-Contrast Porous Media
Eric Chung (The Chinese University of Hong Kong)
In this talk, we present an adaptive physics-preserving multiscale method for incompressible and immiscible two-phase flow in high-contrast porous media. The method couples a physics-preserving implicit pressure explicit saturation scheme (P-IMPES) with a mixed constrained energy minimizing generalized multiscale finite element method. The P-IMPES formulation preserves the conservative structure of the model, treats the wetting and non-wetting phases in an unbiased way, and maintains saturation bounds under a suitable CFL condition. We prove local conservation, unbiased property, bounds preservation, and a saturation error estimate for the multiscale approximation. The work is partially supported by the Hong Kong RGC General Research Fund (Project Nos. 14305423 and 14305624).
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Parallel Finite Element Simulation of the Quasi-Static Biot Model with Application to Brain Edema Modeling
Meng-Huo Chen (National Chung Cheng University)
Brain edema can be modeled by Biot-type poroelastic equations coupling tissue displacement and fluid pressure. We present a parallel finite element framework for the quasi-static Biot model as a step toward large-scale simulations on realistic two- and three-dimensional human brain meshes. The method uses a direct displacement--pressure formulation with Taylor--Hood-type \(P_2\)-\(P_1\) elements and Crank--Nicolson time stepping. The mesh is partitioned by METIS, halo degrees of freedom are constructed for the mixed-order discretization, and the resulting systems are solved by Krylov methods with HYPRE-based preconditioning. Manufactured-solution tests verify the expected convergence rates and reveal solver sensitivity in small-storage and low-permeability regimes. The study also provides a computational basis for future dynamic Biot models, where tissue inertia is included through second-order time derivatives of displacement.
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Locally Conserved and Structure-Preserving Numerical Methods for Flows in Porous Media
Huangxin Chen (Xiamen University)
Modeling and simulation of complex flows in porous media are of great interest in the fields of hydrology and petroleum reservoir engineering. In this talk we will introduce a new locally mass-conserved enriched Petrov-Galerkin (EPG) method for flows in porous media, including the applications of EPG method for flows in fractured porous media. The thermodynamically consistent mathematical models for non-isothermal multi-phase flows in porous media, and the structure-preserving numerical methods will also be discussed.
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A staggered discontinuous Galerkin method for linear elasticity problem on polytopal meshes
Ruishu Wang (Jilin Univercity)
We propose a novel staggered discontinuous Galerkin (SDG) method for linear elasticity, formulated within the Hellinger–Reissner variational principle. The method introduces symmetric stress spaces that ensure normal continuity across element interfaces on arbitrary polytopal meshes, while displacements are approximated using piecewise polynomial functions on the same mesh. The method is locking-free and satisfies a local balance of linear momentum and angular momentum. The formulation also admits a hybridizable structure, which greatly facilitates numerical implementation.
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Locally Conservative Sparse Neural Operator netWorks
Jiyeon Kim (Ajou University)
Operator learning is a promising paradigm for parametric partial differential equations (PDEs), yet network parameters grow rapidly with mesh resolution. We present Sparse Neural Operator netWorks (SNOW), a framework that embeds the sparse structure of PDE discretizations into the network architecture. As a primary application, Sparse FEONet restricts each layer's connectivity to the local support of finite element basis functions, supported by a universal approximation theorem and stability analysis, and achieves up to a 98% parameter reduction while maintaining accuracy. In this talk we propose SNOW-HDG, a locally conservative extension based on the Hybridizable Discontinuous Galerkin (HDG) method. The network learns only the skeleton trace λ using HDG connectivity as its sparsity pattern, and element-wise solutions are recovered by the HDG local solver, so the prediction satisfies local conservation property. Poisson experiments confirm accuracy and efficiency.
MS07 Mathematics and AI for Scientific Computing Main organizer: Yuqing Li (East China Normal University) · 8 talks
Organizers
Main organizer: Yuqing Li (East China Normal University)
- Co-organizer: Xiaofei Zhao (Wuhan University)
- Co-organizer: Zhiping Mao (Eastern Institute of Technology, Ningbo)
Program & Speakers
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A JKO approach for gradient flows
Chaozhen Wei (University of Electronic Science and Technology of China)
In this talk, I will present a novel numerical approach based on minimizing movement (also known as JKO) schemes for a class of gradient flows arising widely in applications in material science and biology such as porous medium, tumor growth, phase separation, crystal defects migration, surface-diffusion-driven dynamics. By leveraging the underlying variational structure and its intrinsic connection to optimal transport, we construct structure-preserving fully discrete JKO scheme that ends up with a series of linearly constrained convex optimization problems, which can be solved by tailored primal dual operator splitting methods. Our method has built-in strictly positivity or bounds preserving, mass conservation, and entropy decreasing properties, and overcomes stability issue due to the strong nonlinearity and degeneracy. I will show a suite of simulation examples to demonstrate the effectiveness and applicability of our approach.
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SPDEBench: A Benchmark for Learning Regular and Singular Stochastic PDEs
Qi Meng (Chinese Academy of Sciences)
Stochastic partial differential equations model rough spatio-temporal phenomena across turbulence, quantum dynamics, materials, and other complex systems, yet machine learning for SPDEs lacks unified evaluation tools that handle stochasticity, discretization, and singular behavior with sufficient care. This keynote introduces SPDEBench, a benchmark for learning both regular and singular SPDEs through controlled data generation, renormalization-aware simulation, representative baselines, and metrics that go beyond pointwise error. We will discuss what SPDEBench reveals about current neural operators and SPDE-aware models, where existing methods succeed or fail, and how principled benchmark design can guide the next generation of learning methods for rough, noisy, and physically significant dynamical systems.
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Randomized Neural Network Methods for Partial Differential Equations
Fei Wang (Xi'an Jiaotong University)
In this talk, we introduce Randomized Neural Network (RaNN) methods as a unified framework that combines random feature-based neural representations with traditional numerical methods for solving partial differential equations. Within this framework, we develop a series of structure-preserving schemes, including RaNN–Petrov–Galerkin, local RaNN–discontinuous Galerkin, local RaNN–hybridizable discontinuous Petrov–Galerkin, and local RaNN–finite difference methods. Adaptive growing RaNN strategies are further developed to enhance approximation capability for complex solution structures, and RaNN-based operator learning approaches are explored for efficient parameterized PDE simulations. Numerical results demonstrate that RaNN methods achieve high accuracy with reduced degrees of freedom while maintaining computational efficiency, structural consistency, and scalability, providing a promising direction for the integration of numerical analysis and machine learning in scientific computing.
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Numerical methods for Dirac equation
Yan Wang (Central China Normal University)
This talk is devoted to numerical methods for highly oscillatory and low-regularity Dirac equations. In the nonrelativistic regime, the solution to Dirac equation is highly oscillatory in time, severe time-step size restriction is required to resolve the solution. With rough initial data, traditional methods usually suffer from order reduction. In this talk, we introduce the operator separation and nested Picard iteration technique to construct efficient numerical methods for this two kinds of problems.
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Deep Onsager Operator Learning: An Unsupervised Framework for Dissipative Equations
Zhenye Wen (Wuhan University)
Most existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons demonstrate its enhanced performance.
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Condensation Phenomena and Stochastic Methods: From Neural Networks to Particle Systems
Yuqing Li (East China Normal University)
We study condensation phenomena driven by stochasticity in neural network training and interacting particle systems. In neural networks, we characterize a phase diagram of condensation in two-layer and convolutional models, where training induces concentration of weights along finitely many directions, improving generalization. We further show that Dropout admits a stochastic modified equation description, which captures its dynamics and explains its role in promoting and stabilizing condensation at late training stages. In a second direction, we develop a quasi-Ewald splitting method for charged particles in nanoscale confinement. By combining importance sampling with random batch techniques in Fourier space, we obtain an O(N) algorithm for electrostatic interactions. The method also reveals dielectric boundary effects and anisotropic diffusion induced by confinement. dimensional structures in learning systems and efficient computation in complex physical systems.
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On action ground state of NLS
Xiaofei Zhao (武汉大学)
In this talk, we first review the definition of action ground state of NLS. Some equivalent formulations are then introduced. Based on each, different computational methods are desigened and compared. With the simulation and analysis, the relation between action and energy ground states are explored.
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NN-WGF: A Neural Ritz Flow-Map Method for Two- and Three-Dimensional Porous Medium Flows
Haozhe Xu (Eastern Institute of Technology, Ningbo)
We present NN-WGF, a neural Ritz discretization of a Lagrangian flow-map variational scheme for Wasserstein gradient flows. Unlike PINNs, the method does not fit PDE residuals or solution data. At each implicit time step, a residual neural network represents the transport map and directly minimizes a weighted movement cost plus the porous-medium internal energy. The density is reconstructed by rho(T(X),t)=rho_0(X)/det(DT(X)), so discrete Lagrangian mass is conserved by construction. We evaluate the method on the compactly supported Barenblatt benchmark with m=5, dt=0.01, and 400 steps to T=4. Against the exact self-similar scaling, the final relative density errors are 1.94% in 2D and 1.72% in 3D, with map errors of 0.40% and 0.55%. Both runs exhibit energy dissipation, zero reported mass drift, and positive sampled Jacobians. The results support neural flow-map optimization for long-time free-boundary Wasserstein dynamics in three dimensions.
MS08 Recent advances in analysis and numerical methods for wave scattering problems Main organizer: Tao Yin (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) · 6 talks
Organizers
Main organizer: Tao Yin (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Co-organizer: Weiying Zheng (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Program & Speakers
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Wave-number-explicit analysis for maxwell's equation with Dirichlet-to-Neumann truncation
Xue Jiang (Beijing University of Technology)
This work is focused on the propagation of electromagnetic waves in R^3 described by Maxwell's equation with large wave number and Silver-Muller radiation condition. The model problem is approximated by truncating the exact Dirichlet-to-Neumann (DtN) operator into a finite sum of vector spherical harmonics. We prove the well-posedness and wave-number-explicit H(curl)-stability of the solution to truncated problem by assuming that the truncation number N satisfies N≥akR for some a> 1, where k represents the wave number and R is the radius of the physical domain. Additionally, we demonstrate that the truncated solution is exponentially close, in terms of N, to the true scattering solution. Finally, we present the hp-finite element method (hp-FEM) for the truncated problem, along with its asymptotic error estimate. Some numerical experiments are provided to validate the theoretical findings.
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Nanoscale electromagnetic scattering: well-posedness and adaptive FEM
Tao Yin (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
This talk will present our recent work on the mathematical analysis and numerical method for the nanoscale electromagnetic scattering problem with mesoscopic boundary conditions whose TM polarization leads to a transmission problem of Helmholtz equations with boundary conditions involving the Laplace-Beltrami surface differential operator. In particular, a nonstandard Sobolev space involving the jumps across the boundary is introduced to establish the well-posedness of the variational formulations for the reduced problems truncated by the Helmholtz Dirichlet-to-Neumann operator. The a priori and a posteriori error estimates of the finite element discretization are derived by means of some carefully designed local error estimators that are related to the mesoscopic boundary conditions. Numerical experiments will be presented to validate the theoretical results and verify the convergence and efficiency of the proposed adaptive method.
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Wave-number-explicit stability and preasymptotic error analysis of a UPML finite element method
聿昊 王 (中科院数学与系统科学研究院)
This talk presents a wavenumber-explicit stability and preasymptotic error analysis for the two-dimensional Helmholtz exterior scattering problem with a uniaxial perfectly matched layer (UPML) truncation. The UPML formulation is obtained by Cartesian complex stretching, and the analysis relies on estimates of the stretched Green kernel. We prove stability estimates for the whole-space and truncated UPML problems. In particular, the truncated formulation has an inf-sup constant bounded below by $Ck^{-1}$ in a natural $k$-weighted $H^1$ norm, equivalently a stability bound of order $O(k)$. We also obtain exponential decay of the PML truncation error. Based on this framework, we study a linear CIP-FEM. A key ingredient is a piecewise $H^{1+s}$-regularity result for any $0<s<1$, accounting for coefficient jumps and corner singularities. This yields wavenumber-explicit preasymptotic error estimates, illustrated by numerical experiments.
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A Robust Multigrid Method for the High Wavenumber Helmholtz Equation via Ray Equation Correction
Yicheng Huang (Academic of Mathematics and System Science, Chinese Academy of Science)
Solving the high-wavenumber Helmholtz equation with impedance boundary conditions remains a computational challenge due to the indefiniteness of the resulting discrete systems. In this work, we propose an efficient multigrid framework designed to achieve robust, weakly wavenumber-dependent convergence. Our approach utilizes a ray equation correction, $u=a\mathrm{e}^{\mathbf{i}\bm{w}\cdot\bm{x}}$ to reformulate the original problem into a complex-coefficient convection-diffusion equation for the amplitude variable a. We solve this transformed system using a specialized geometric multigrid algorithm. Theoretical analysis and numerical experiments confirm that our methodology maintains optimal $\mathcal{O}(N)$ computational complexity, with iteration counts exhibiting slow growth as the wavenumber k increases.
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Structure-Preserving and Helicity-Conserving FEMs For MHD Systems
Shipeng Mao (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Designing numerical methods that accurately preserve mathematical structures and physical properties has become a critical research topic in MHD computations. For incompressible MHD, we propose the fully discrete finite element exterior calculus (FEEC) method that simultaneously and exactly preserves key physical properties-including mass conservation, magnetic flux conservation, current density conservation, energy conservation, and the conservation of magnetic helicity and fluid helicity-even under their respective physical limits. For the ideal compressible MHD system, we 4y develop a Lagrangian FEEC algorithm that achieves high-order accuracy in both space and time, preserves positivity of density, ensures a divergence-free magnetic field, and conserves magnetic helicity. To address mesh distortion issues, we design and develop a structure-preserving, helicity-conserving ALE FEEC algorithm.
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The PML-BIE method: numerics and theories
Wangtao Lu (Zhejiang University)
In this talk, I will briefly review the progresses of the PML-BIE method for wave scattering problems, and then will talk about the convergence theory of such method for a particular scattering problem.
MS09 Advances in Computational Mathematics and Machine Learning for Scientific Computing Main organizer: BingZe Lu (Department of mathematics, National Chung Cheng University) · 7 talks
Organizers
Main organizer: BingZe Lu (Department of mathematics, National Chung Cheng University)
- Co-organizer: Yu-Chen Shu (Department of Mathematics, National Cheng Kung University, Taiwan)
Program & Speakers
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Convergence Theory for Monotone Finite-Difference Schemes Solving Prescribed Jacobian Equations on Unbounded Subsets of $\mathbb{R}^n$
Axel Turnquist (Beijing Institute of Mathematical Sciences and Applications)
Consider two probability measures $\mu$ and $\nu$ in $\mathbb{R}^n$ with unbounded supports and with density functions $f,g$ and a pushforward map $T$ such that $T_{\#}\mu = \nu$. If $T$ is smooth enough, it satisfies the Jacobian equation $f(x) = g(T(x)) \vert \det DT(x) \vert$. In many applications, $T$ is related to the gradient of a so-called potential function $u$. In many cases of optimal transport and freeform optics problems, $T(x) = F(x, u(x), D u(x))$. In this case, the PDE for $u$ is called a prescribed Jacobian equation (PJE). We aim to solve PJEs with monotone finite-difference methods on unbounded subsets of $\mathbb{R}^n$, where we cut off the problem at a radius $R>0$. We present explicit convergence rates for the potential function in the quadratic cost optimal transport case as $R \rightarrow \infty$. We then present results on the convergence of monotone discretizations for PJEs, given that the potential function $u$ is locally Lipschitz or H\"{o}lder continuous.
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Particle trajectories in 1D Boussinesq equation
JUANMING YUAN (Providence University)
In this work, we consider particle trajectories beneath soliton solutions described by the Boussinesq equation, a model for bidirectional, weakly nonlinear, long surface waves. The Boussinesq equation admits many exact soliton solutions. We analyze the particle drift induced by both single-soliton and various two-soliton solutions. Our results provide insight into the Lagrangian behavior of fluid particles under these nonlinear wave fields.
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Artifacts of Numerical Integration in Learning Dynamical Systems
Bingze Lu (National Chung Cheng University)
When learning a dynamical system from data, you fit a model by minimizing mismatch with observed trajectories — but this requires a numerical integrator, and the choice of integrator shapes what you learn. This talk shows the integrator can corrupt identified dynamics: a damped oscillation can be "learned" as anti-damped and oscillating in the wrong direction, yet still fit the data perfectly. The culprit is the integrator's stability region, which distorts which dynamics appear consistent with observations. The obvious fixes don't help: smaller step sizes or higher-order explicit methods make things worse, since higher-order explicit integrators have larger stability regions reaching further into the "wrong" part of the complex plane. The remedy: the implicit midpoint method, which provably preserves conservative or dissipative structure from discrete data — a principled default even when the only assumption is that the system is autonomous.
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Conformal Symplectic Neural Operators for Hamiltonian Partial Differential Equations in Complex Domains
Takaharu Yaguchi (Kobe University)
Hamiltonian partial differential equations include the wave equation and the Maxwell equations, and typically describe wave phenomena. To efficiently compute solutions to such equations on complex domains, some techniques for handling domain geometry are required. The Brinkman method is an example of such methods. In this talk, we explain that equations obtained by applying the Brinkman method to the Hamilton partial differential equations possess a conformal symplectic structure. Based on this fact, we propose an operator learning method that preserves this structure.
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Neural Initialization and Local Search for Accelerating Digital and GPU Annealers in QUBO Problems
Yu-Chen Shu (National Cheng Kung University)
Quadratic unconstrained binary optimization (QUBO) is a common framework for modeling large-scale combinatorial optimization problems in scientific computing, scheduling, manufacturing, and resource allocation. This talk presents a neural initialization and local search strategy for accelerating QUBO solvers based on digital and GPU annealers. From each QUBO matrix, bit-wise structural features are extracted and used by a neural network to predict variable-wise probabilities. Candidate binary solutions are generated by thresholding and Bernoulli sampling, refined by local search, and used as warm-starts for annealing. Experiments on random QUBO matrices and synthetic subset-sum instances show that the proposed approach reduces solve time while maintaining final objective quality.
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Multivariate Müntz–Szász Neural Networks for Learning with Adaptive Power-Law Bases
Ming Cheng Shiue (National Yang Ming Chiao Tung University)
Building upon the Müntz–Szász Network (MSN) in the literature for univariate function approximation, we propose Multivariate Müntz–Szász Neural Networks (MMSNs) for learning high-dimensional functions. The proposed architecture employs adaptive multivariate power-law basis functions with trainable exponents, providing an efficient and interpretable representation of multiscale and anisotropic features. We present the mathematical foundation of MMSNs together with numerical experiments demonstrating their approximation accuracy and effectiveness for high-dimensional function learning.
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Finite-dimensional approximation of push-forwards via jets and data
Isao Ishikawa (Kyoto University)
Jets are canonical local objects, and they provide finite-dimensional models for operators induced by analytic maps, in particular push-forwards. In this talk, I will explain how this yields finite-dimensional approximations of push-forwards, explicit error bounds, and reconstruction of analytic vector fields from data from a dynamical system. This provides a novel mathematical framework for data-driven analysis of dynamical systems, including Koopman-type approximations from sampled data. The main point is that approximation and reconstruction both arise from the same jet structure. I will describe how local jet information can be used to build finite-dimensional models, quantify their accuracy, and recover analytic dynamics from observations. Time permitting, I will briefly comment on how the same viewpoint also leads to local dynamical restrictions through operator-theoretic properties.
MS10 Modeling Health and Disease in the Age of AI Main organizer: Eunha Shim (Department of Mathematics, Soongsil University) · 3 talks
Organizers
Main organizer: Eunha Shim (Department of Mathematics, Soongsil University)
Program & Speakers
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Cost-effectiveness of female-only and gender-neutral HPV vaccination strategies in Japan
Minjin Kim (Soongsil university)
This study evaluated the cost-effectiveness of female-only vaccination (FOV) and gender-neutral vaccination (GNV) strategies using 4-valent (4vHPV) and 9-valent (9vHPV) HPV vaccines among Japanese adolescents aged 12–16 years. An age- and sex-structured dynamic transmission model simulated HPV transmission and disease progression over a 100-year period. Six strategies were assessed: no further vaccination; FOV with 4vHPV; FOV with 9vHPV; GNV with 9vHPV for females and 4vHPV for males; GNV with 4vHPV for both sexes; and GNV with 9vHPV for both sexes. Both FOV with 4vHPV and FOV with 9vHPV produced ICERs below the willingness-to-pay threshold. GNV with 9vHPV for both sexes generated the greatest health benefits while remaining cost-effective. These findings indicate that both female-only 4vHPV and 9vHPV vaccination strategies are cost-effective under Japan's economic standards, while GNV with 9vHPV offers greater health gains and overall cost-effectiveness.
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Statistical Hazard Models for Mendelian Randomization in Large-scale Survival Data
Wonil Chung (Soongsil University)
Mendelian randomization (MR) with survival outcomes commonly relies on the Cox proportional hazards (CoxPH) model, despite limitations arising from the proportional hazards assumption and the non-collapsibility of the hazard ratio. This paper compares CoxPH and additive hazards models for survival-based MR in terms of causal interpretation, estimator behavior, and model adequacy. We develop a simulation framework based on restricted mean survival time (RMST) to enable fair comparison across hazard structures. When the hazard model is correctly specified, CoxPH is more efficient for multiplicative hazards, whereas additive hazards provide stable inference on a collapsible scale; under misspecification, CoxPH exhibits systematic bias, while additive hazards generally maintain smaller bias at the cost of higher variance. An application to colorectal cancer survival highlights heterogeneity in hazard-model suitability.
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Analysis of longitudinal antibody titers measured after COVID-19 mRNA vaccination
Hyeongki Park (Pusan National University)
A key challenge in the post–COVID-19 era is optimizing repeated vaccination against circulating and emerging SARS-CoV-2 variants while efficiently managing medical resources. Identifying poor responders with low sustained antibody titers may help prioritize revaccination. We analyzed longitudinal antibody titer data from 2,526 participants in Fukushima, Japan, collected between April 2021 and November 2022. Using mathematical modeling and machine learning, we stratified antibody trajectories after two primary doses and one booster dose of mRNA vaccines into three groups: durable, vulnerable, and rapid-decliner. The rapid-decliner group experienced earlier infections than the others. In addition, early post-booster spike-specific IgA titers were lower in participants who later developed SARS-CoV-2 infection. This framework may help guide vaccine distribution strategies to maximize population-level immunity.
MS11 Statistical Learning for Inverse Problem: Methodologies and Applications Main organizer: Hongqiao Wang (Central South University) · 5 talks
Organizers
Main organizer: Hongqiao Wang (Central South University)
- Co-organizer: Qiuqi Li (Hunan University)
- Co-organizer: Xiang Sun (Ocean University of China)
Program & Speakers
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Tensor decomposition-based neural operator with dynamic mode decomposition for parameterized time-dependent problems
Yuanhong Chen (Qingdao University of Science and Technology)
DeepONets have gained attention for modeling PDEs but often fail to accurately predict solutions outside the training time interval due to poor extrapolation. To address this, we propose TDMD-DeepONet, a novel framework combining tensor train decomposition (TTD) and dynamic mode decomposition (DMD). We first show the mathematical consistency between TTD and DeepONet. TTD is applied to a higher-order tensor of spatiotemporal snapshots across parameter values, yielding parameter-, space-, and time-dependent cores. DMD then models the evolution of the time core, which together with the space cores forms the trunk net. The branch net is a neural network taking parameters as inputs, and its outputs are merged with the trunk net for prediction. We further propose ETDMD-DeepONet, which adds a linear layer to the branch net, taking projected initial conditions as input. Numerical examples show that both proposed methods outperform standard DeepONet in forecasting accuracy.
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A sensitivity analysis-based variable-separation method for eigenvalue problems with random inputs
YUMING BA (Guangdong Polytechnic Normal University)
In this presentation, we present a Variable-separation (VS) method based on sensitivity analysis for solving eigenvalue problems with random inputs. The method employs an offline-online strategy. During the offline stage, deterministic functions are precomputed for a training set and stored, then repeatedly used for parametric eigenvalue problems (PEVPs) with new parameters. In applications, small parameter variations can cause dramatic changes in the eigenvalues and eigenvectors. Therefore, a sensitivity analysis is essential to ensure its effectiveness. For low-sensitive regions, only a single global VS model is required. For highly sensitive regions, the parameter domain is divided into several subregions, and multiple local VS models are constructed. Furthermore, the Grassmann distance of eigenspaces is used to evaluate the performance of the proposed method. Finally, we apply the proposed method to a parametric matrix based on Karhunen–Loève expansion.
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Data-adaptive RKHS regularization for learning convolution kernels
Haibo Li (Huazhong University of Science and Technology)
Learning convolution kernels in operators from data arises in numerous applications and represents an ill-posed inverse problem. In this talk, I will present a Data-Adaptive RKHS (DA-RKHS) regularization framework for learning convolution kernels from data. We show that the input data and forward operator themselves induce a data-adaptive RKHS, and there is a finite set of automatic basis functions to represent the estimators in the DA-RKHS when the observation data is discrete and finite. We propose Tikhonov regularization, iterative regularization, and divide-and-conquer methods for small, large, and huge data, respectively. Theoretical results show that the DA-RKHS estimator can achieved optimal minimax convergence rate under certain source conditions. Numerical experiments confirm that our automatic RKHS regularization outperforms standard Gaussian process methods with preselected kernels.
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Non-Intrusive POD–ME-gPC Reduced-Order Modeling for Parameterized Nonlinear Dynamical Systems
Xiaomin Wang (Yonsei University)
We propose a non-intrusive reduced-order modelling framework for uncertainty quantification and surrogate evaluation in parameterized nonlinear systems. The method combines two-stage Proper Orthogonal Decomposition (POD) with adaptive Multi-Element generalized Polynomial Chaos (ME-gPC). POD extracts low-dimensional spatial and temporal structures from full-order snapshots, while ME-gPC approximates the parameter-dependent coefficients by local polynomial expansions on an adaptively partitioned stochastic space. This avoids the global smoothness assumption of standard POD–PCE and improves robustness for parameter-sensitive dynamics. The framework is tested on the Kraichnan–Orszag system, stochastic advection–diffusion, flow past a circular cylinder, and a flexible flapping filament. Results show accurate reconstruction, long-time prediction, and uncertainty bands at unseen parameters, indicating its potential for repeated forward solves in statistical inverse problems.
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Recovery of pairwise interaction kernels from noisy position trajectories with structure-aware neural ODEs
Jinchao Feng (Great Bay University)
Recovering pairwise interaction kernels from particle trajectories is a central inverse problem in computational physics. It arises in molecular dynamics, soft matter, coupled oscillators, and collective-motion modeling. Existing kernel-learning methods often rely on velocity data, which may be unavailable and must then be approximated from noisy position observations. However, such finite-difference approximations strongly amplify noise under sparse temporal sampling. To overcome this limitation, we propose a structure-aware Neural ODE framework for learning pairwise interaction kernels directly from noisy trajectory data. We also establish well-posedness of the inverse problem and provide an error estimate for the learned kernel. We test our method on different types of systems and on a real pedestrian-trajectory dataset. All examples show improved robustness over finite-difference baselines on noisy annotated data.
MS12 Numerical PDEs for scientific machine learning and the vice versa Main organizer: Hyea Hyun Kim (Kyung Hee University) · 8 talks
Organizers
Main organizer: Hyea Hyun Kim (Kyung Hee University)
- Co-organizer: Eric Chung (The Chinese University of Hong Kong)
Program & Speakers
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Numerics-Guided Scientific Machine Learning
Youngjoon Hong (Seoul National University)
Scientific machine learning provides new tools for solving partial differential equations, but reliable performance often requires incorporating mathematical and numerical structures into the learning process. In this talk, I will discuss numerics-guided SciML methods for challenging PDEs, focusing on approaches that combine neural network-based coefficient learning with classical numerical trial spaces. I will present examples inspired by finite element, discontinuous Galerkin, spectral element, and Müntz-type element methods. These methods exploit problem-adapted approximation spaces and numerical stability mechanisms, making them effective for PDEs with fluid flow, boundary layers, multiscale behavior, discontinuities, or singular perturbations. I will also discuss how numerical analysis helps explain their accuracy, stability, and generalization.
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Domain decomposition based iterative algorithms for neural network approximation to PDEs
Hyea Hyun Kim (Kyung Hee University)
Domain decomposition methods (DDMs) have been extensively developed for fast solutions of algebraic systems obtained from classical finite element discretization of partial differential equations (PDEs). The problem domain is partitioned into overlapping or nonoverlapping subdomains and each local problem to each subdomain is solved iteratively with a suitable interface update to obtain convergent local solutions to the global solution. Each local problem can be solved in parallel and with the addition of a global coarse problem, the iteration convergence can become robust to the number of subdomains in the partition. Recently, DDMs have been also applied to neural network approximation to PDEs in order to improve the solution accuracy and training efficiency. In this talk, Robin iterative algorithms for neural network approximation to PDEs are presented with their convergence theory and numerical results.
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Hybrid Least Squares/Gradient Descent Methods for DeepONets
Chang-Ock Lee (KAIST)
We propose an efficient hybrid least squares/gradient descent method to accelerate DeepONet training. Since the output of DeepONet can be viewed as linear with respect to the last layer parameters of the branch network, these parameters can be optimized using a least squares (LS) solve, and the remaining hidden layer parameters are updated by means of gradient descent form. However, building the LS system for all possible combinations of branch and trunk inputs yields a prohibitively large linear problem that is infeasible to solve directly. To address this issue, our method decomposes the large LS system into two smaller, more manageable subproblems - one for the branch network and one for the trunk network - and solves them separately. This method is generalized to a broader type of L^2 loss with a regularization term for the last layer parameters, including the case of unsupervised learning with physics-informed loss.
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Asymptotic Expansion Theory and Methods for Multiscale Hyperelastic Problems
Jizu Huang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
In this talk, we study the theory and methods of multiscale asymptotic expansions for hyperelastic problems. Based on the multiscale asymptotic expansion framework, we propose a homogenization method and a first-order asymptotic expansion method for hyperelastic problems. We prove the convergence of the energies of the homogenized solution and the first-order asymptotic approximation. Numerical experiments further verify the accuracy and effectiveness of the proposed methods.
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Physics-Informed Neural Network Approximations for the Allen-Cahn Equation Using Temporal Decomposition and Levenberg-Marquardt Optimization
Seokjun Ham (Kyung Hee University)
We propose temporal decomposition strategies for improving the accuracy of physics-informed neural network (PINN) solutions for the Allen-Cahn equation, which is characterized by stiff nonlinearity and sharp interface dynamics. The proposed approaches combine temporal decomposition with the Levenberg-Marquardt (LM) optimization algorithm to enhance the convergence and efficiency of PINN training. Three temporal decomposition strategies are considered: sequential time marching, a time-window approach, and a semi-discrete formulation based on the backward differentiation formula (BDF). Numerical results show that the three strategies achieve comparable accuracy when combined with the LM optimizer.
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Physics-Informed Neural Networks for Multiscale PDEs with Discontinuous High-Contrast Coefficients
Wing Tat Leung (City university of Hong Kong)
Physics-informed neural networks (PINNs) are data-driven approaches to solving equations. It is successful in many applications; however, the accuracy of the PINN is not satisfactory when it is used to solve multiscale equations. In this paper, we propose a PINN to solve PDEs with a discontinuous high-contrast coefficient. To find such a solution using a neural network representation, we introduce an additional feature input to the network to retain the inherent solution properties. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation. Using the specially designed residual, we can handle the discontinuity of the high-contrast coefficient. Numerical results show that even with a one-hidden-layer (shallow) network and a moderate number of neurons, the present network model can achieve an accurate result using sufficient training data points.
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Learning coarse spaces for Darcy flow in heterogeneous media
Eric Chung (The Chinese University of Hong Kong)
In this talk, we present Darcy flow problems with random permeability using iterative solvers, enhanced by a two-level preconditioner based on a generalized multiscale coarse spaces, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedite the construction of the generalized multiscale coarse spaces. Comprehensive numerical experiments involving diverse random coefficient models, benchmarked against standard algebraic multigrid and ILU preconditioners, demonstrate that the proposed approach substantially reduces the time required to generate the prolongation operator while preserving the preconditioner's original efficacy over traditional methods. The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Projects: 14305624 and 14305423).
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Power Contractivity and $\mathcal{O}(k)$ Solvers for High-Frequency Helmholtz and Maxwell Problems
Shihua Gong (The Chinese University of Hong Kong, Shenzhen)
High-frequency Helmholtz and Maxwell discretizations produce large, non-Hermitian, and highly indefinite linear systems. This talk presents a power-contractivity convergence theory for overlapping Schwarz methods and its application to scalable frequency-domain wave solvers. For Helmholtz problems, we analyze the power contractivity of RAS-Imp and RAS-PML, leading to a massively parallel RAS-PML solver that exhibits approximately $\mathcal{O}(k)$ runtime for problems with $\mathcal{O}(k^d)$ unknowns. For time-harmonic Maxwell equations, we establish the well-posedness of the parallel overlapping Schwarz iteration, characterize error propagation through impedance-to-impedance maps, and derive a power-contractivity criterion for RAS-Imp. Numerical experiments illustrate the predicted power-contractivity behavior and provide preliminary evidence for the effectiveness of RAS-Imp and RAS-PML for Maxwell problems.
MS13 Numerical Optimization and Related Applications Main organizer: Mei-Heng Yueh (National Taiwan Normal University) · 4 talks
Organizers
Main organizer: Mei-Heng Yueh (National Taiwan Normal University)
Program & Speakers
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Nodal Newton-Noda Iteration for Computing Eigenstates of Nonlinear Schrödinger Equations
Ching-Sung Liu (National University of Kaohsiung)
We introduce a novel Nodal Newton-Noda Iteration (N-NNI) for efficiently computing the eigenstates of one-dimensional nonlinear Schrödinger equations. Since the $K$-th state changes sign exactly $K$ times, partitioning the domain at these nodes splits the problem into $K+1$ positive nonlinear algebraic eigenvalue problems (NAEPs); the case $K=0$ gives the ground state. N-NNI employs a two-level iterative structure: an outer iteration that refines the nodal partition via a global Newton step, and an inner iteration that solves the positive subproblems by block-elimination of the coupled Newton system. A key feature of N-NNI is that the subdomain iterates stay strictly positive while the eigenvalue sequence increases monotonically and converges quadratically. The dominant computational cost lies in solving the Newton systems in the inner iterations. Numerical results demonstrate the efficiency and reliability of the proposed method, supporting our theoretical findings.
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The Minimal Solution of Positive Semidefinite Quadratic Programming Problems
Yueh-Cheng Kuo (National Chengchi University)
We study convex quadratic programming problems with positive semidefinite Hessian matrices, for which the set of optimal solutions may be nonunique. We prove that, when the optimal value is finite, there exists a unique minimal solution, defined as the minimizer with the smallest Euclidean norm, and provide a primal-dual characterization of this solution. To compute the minimal solution efficiently, we introduce a regularization framework that yields a sequence of strictly convex quadratic programs and prove that their solutions converge to the minimal solution of the original problem. Based on this analysis, we develop an active-set-based algorithm with a homotopy strategy for tracking the regularization parameter. Numerical experiments demonstrate the effectiveness and robustness of the proposed method.
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An optimal parameterized Newton-type structure-preserving doubling algorithm for finite-time state-dependent Riccati equation
Tsung-Ming Huang (National Taiwan Normal University)
This study develops a high-performance finite-time state-dependent Riccati equation (FT-SDRE) algorithm integrated with structure-preserving doubling algorithms (SDAs). At each state of the FT-SDRE, a modified Newton–Lyapunov method is employed to solve the continuous algebraic Riccati equation (CARE). Furthermore, we introduce a simplified SDA with adaptive optimal parameter selection to efficiently solve the associated Lyapunov equation. Numerical results demonstrate that our accelerated FT-SDRE algorithm is approximately three times faster than standard implementations using MATLAB’s \textbf{icare} and \textbf{lyap} functions.
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Numerical Optimization for Spherical Area-Preserving Parameterizations
Mei-Heng Yueh (National Taiwan Normal University)
Area-preserving surface parameterization is a fundamental problem in computational geometry, with applications to surface registration and shape analysis. In this talk, I will present a numerical optimization framework for computing spherical area-preserving parameterizations of genus-zero closed surfaces. The proposed framework is based on minimizing a spherical authalic energy and introduces a Riemannian bijective correction to guarantee bijective parameterizations. Numerical examples and applications to surface shape analysis will be presented.
MS14 New developments in numerical verification and analysis for nonlinear problems Main organizer: Tomoyuki Miyaji (Kyoto University) · 7 talks
Organizers
Main organizer: Tomoyuki Miyaji (Kyoto University)
Program & Speakers
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Numerical verification of unimodal solutions of the generalized Constantin--Lax--Majda equation with viscosity
Tomoyuki Miyaji (Kyoto University)
We present a computer-assisted proof of the existence of unimodal solutions of the generalized Constantin--Lax--Majda (gCLM) equation with viscosity and an external force, which is regarded as a one-dimensional model of two-dimensional fluid flow. In the numerical study by Kim et al. (Jpn. J. Ind. Appl. Math. 35 (2018) 1065--1083), it was found that the equation possesses unimodal solutions at sufficiently high Reynolds numbers. In this model, a solution and its second antiderivative are identified with the vorticity and the stream function up to sign, respectively. The solution is said to be unimodal the stream function has exactly one maximum and one minimum in the interval. We apply a fixed-point theorem in infinite-dimensional spaces and interval arithmetic to prove the existence of solutions. This talk is based on joint work with Prof. Yoshitaka Watanabe (Kyushu University).
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Numerical Verification and Analysis towards Jeffery-Hamel flows related to Navier-Stokes equations
Yoshitaka Watanabe (Kyushu University)
Jeffery-Hamel flows are exact solutions of the Navier-Stokes equations in a two-dimensional domain bounded by two semi-infinite lines that meet at a point. It is known that each solution of Jeffery-Hamel flows can be represented in terms of elliptic functions. Hamel's original paper explored a wider class of solutions, and describing all of these solutions in terms of elliptic functions is difficult. We will numerically verify Jeffery-Hamel flows. This approach will avoid the use of elliptic functions and allow extensions to broader solution classes. In this talk, we will introduce Jeffry-Hamel flows and outline future issues. We also plan to discuss new verification examples.
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Energy dissipation in numerical computation for gradient flows
Yuki Ueda (Hokkaido University)
In this talk, we present a numerical scheme for $H^{-1}$ gradient flows which is based on the backward Euler temporal discretization. The scheme can be interpreted as a minimization problem, and regarded as the minimizing movement with discretized norm. The approximate solutions inherit the energy dissipation property of the continuous system, and this structure can be expected to provide a natural connection to numerical verification. We review the mathematical background of this formulation and demonstrate the application of numerical techniques for solving the associated minimization problems.
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Validated Numerics for ODEs with Conserved Quantities: Application to Black Hole Orbits
Rei Iwamoto (The University of Electro-Communications)
Numerical verification methods for ODEs compute rigorous enclosures that are guaranteed to contain the true solutions. In conservative systems, however, such methods may suffer from severe overestimation (the wrapping effect), while conserved-quantity constraints are not explicitly used. We propose a validated integration method that uses multiple first integrals to reduce such overestimation. At each step, we construct a frame that separates directions constrained by the first integrals from directions along the motion. In this frame, we remove regions violating the conservation laws by combining interval reduction and linear prediction. For geodesic motion in Schwarzschild spacetime (a strongly nonlinear system with three invariants), the proposed method extends the validated time horizon by a factor of approximately 21 (from proper time 17,941 to 376,579) and, in high-precision regimes, outperforms standard methods in both accuracy and computational efficiency.
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The Surface-Constrained Level Set Method for Multiphase Interfacial Motions
Elliott Ginder (Meiji University)
The level set method is one of the most power tools of the applied mathematician. It allows one to mathematically express the evolution of shapes and is used throughout the sciences to model phenomena ranging from the free-surface motion of fluids, to the processing of digital images. Nevertheless, the level set method has largely been limited to modeling the evolution of shapes in euclidean spaces. This research extends the level set method to the multiphase surface-constrained setting where surfaces are defined by point clouds.
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Sharp Dirichlet Eigenvalue Inequalities on Triangles
Ryoki Endo (Niigata University)
This study proves sharp inequalities for the first Dirichlet eigenvalue of the Laplacian on planar triangles. We settle a conjecture of Siudeja on an optimal two-term lower bound involving the area and perimeter, or equivalently a scale-invariant minimization problem formulated by Laugesen and Siudeja. We also prove a Cheeger-type inequality for triangles with the sharp constant conjectured by Parini. The proof is based on a computer-assisted method. We introduce a computable lower bound for second-order directional derivatives of Dirichlet eigenvalues under vertex perturbations. This is combined with validated finite-element estimates, interval arithmetic, and analytic estimates for nearly degenerate triangles.
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A priori error estimates for finite element solutions of the Poisson equation on non-convex domains using graded meshes
Kenta Kobayashi (Hitotsubashi University)
We derive an a priori error estimate with explicit error bounds for finite element solutions of the Poisson equation on non-convex domains. On non-convex domains, the reduced regularity of the solution generally prevents the optimal O(h) convergence rate in the H^1-norm from being achieved with a uniform mesh. To overcome this difficulty, we employ a graded mesh. Unlike conventional graded meshes, which typically require more than O(h^{-2}) elements, the graded mesh proposed in our research uses only O(h^{-2}) elements, the same order as a uniform mesh, while still achieving the optimal O(h) convergence rate.
MS15 Numerical Methods for Semiconductor Device Simulation Main organizer: Donghang Zhang (Academy of Mathematics and System Science, Chinese Academy of Sciences) · 6 talks
Organizers
Main organizer: Donghang Zhang (Academy of Mathematics and System Science, Chinese Academy of Sciences)
- Co-organizer: Weiying Zheng (Academy of Mathematics and System Science, Chinese Academy of Sciences)
Program & Speakers
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Multiphysics Simulation and Efficient Numerical Methods for Semiconductor Devices and Circuit Structures
Tao Cui (Chinese Academy of Sciences)
As semiconductor devices scale toward three-dimensional and stacked architectures, multiphysics simulation has become increasingly important for understanding device behavior, reliability, and device-circuit interactions. This talk presents our recent progress in multiphysics simulation of semiconductor devices and basic circuit structures, including inverters and ring oscillators. The presentation covers stable and efficient finite-element discretization methods for macroscopic transport, quantum transport, and multiscale coupled transport models, as well as numerical methods for the coupled drift-diffusion and conductor equations used in unified device-interconnect simulation. The talk will also introduce the latest developments of DESIGN (DEvice SImulation Gallery at Nanoscale), a semiconductor device simulation framework developed on the PHG (Parallel Hierarchical Grid) platform for large-scale and high-performance parallel computing.
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Quantum Preconditioning Algorithms for Large Linear Systems via Schrödingerization
Chuwen Ma (East China Normal University)
We present a quantum computational framework for solving linear algebra systems arising from iterative solvers. The framework combines the Schrödingerization technique [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett. 133, 230602 (2024)] with preconditioning strategies to achieve near-optimal complexity. Schrödingerization transforms linear differential equations into Schrödinger-type systems with unitary evolution in a higher dimension, making them suitable for quantum computation. In particular, the BPX preconditioner is integrated with Schrödingerization to obtain near-optimal complexity for elliptic problems. Building on this, we extend the method to the Helmholtz equation, incorporating dispersion correction and tailored preconditioning to efficiently address indefiniteness. These results show that quantum preconditioning provides a scalable pathway from well-conditioned systems to challenging wave problems.
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p-multigrid method for high-order discontinuous Galerkin discretization of elliptic problems
Donghang Zhang (Chinese Academy of Sciences)
In this talk, we develop a W-cycle p-multigrid method for elliptic problems, and further extend it to solve elliptic problems with discontinuous coefficients. It adopts hierarchical Legendre bases for SIPDG and SWIPDG discretization. We present unified convergence analysis for this multigrid method. The analysis includes inherited and non-inherited bilinear forms for SIPDG discretization, and relies on piecewise H^2-regularity for SWIPDG discretization with discontinuous coefficients. The proposed method lowers smoothing complexity from traditional O(p^2) to O(p), where p denotes the polynomial degree. It achieves uniform convergence independent of mesh size, polynomial degree and coefficient jump ratio, showing strong robustness for problems with large coefficient jumps. Numerical experiments validate all theoretical results. Its performance is also verified for elliptic interface problems on C^2-smooth interfaces via unfitted finite element discretization.
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On The Stationary Electrothermal Drift-Diffusion Model
WENHAO LU (National University of Singapore)
A generalized bipolar electrothermal drift-diffusion model for semiconductor devices is considered. The model couples the electron and hole continuity equations with the Poisson equation for the electrostatic potential and a heat equation whose source terms include Joule heating and heat generated by recombination. The talk will discuss the mathematical well-posedness of the model, including the existence of weak solutions in a general setting with temperature-dependent mobilities that may degenerate at high temperatures. The existence proof is based on a Schauder fixed-point argument, while uniqueness is established under a suitable low-bias condition. Numerical experiments will also be presented to illustrate the coupled electrothermal effects, the influence of temperature-dependent mobilities, and the behavior of the model under different applied bias conditions.
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Structure preserving schemes for PNP equations
Yongyong Cai (Beijing Normal University)
We present a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Rigorous error estimates in $L^2$ norm are established, which are both second order accurate in space and time. For extensions to the time-independent problem, we construct a HDG scheme for real-world device applications.
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Numerical discretization of a phase field model in ferroelectric devices
Xiaodi Zhang (Zhengzhou University)
In this talk, we present an energy stable scheme for a phase field model in ferroelectric devices. The model is a nonlinear and multi-domain coupled system, which couples the time-dependent Ginzburg-Landau equation for the polarization and Poisson equation for the electric potential. We first deduce the energy estimates for the continuous system and then develop a semi-discrete scheme based on the idea of convex-splitting. The scheme is proved to be unconditionally energy stable and uniquely solvable. Drawing upon the energy argument, error estimates for all variables are established. Several numerical examples including polarization-electric field hysteresis and domain wall dynamics are provided to confirm our theoretical analysis and demonstrate the effectiveness of the proposed method.
MS16 Recent Advances in Numerical Shape and Topology Optimization Methods: From Theory to Practice Main organizer: Wei Gong (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) · 6 talks
Organizers
Main organizer: Wei Gong (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Co-organizer: Julius Fergy Rabago (Kanazawa University, Japan)
- Co-organizer: Shengfeng Zhu (East China Normal University, China)
Program & Speakers
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Adaptive FEM for the Borrvall-Petersson model in topology optimization of Stokes flow
Yifeng Xu (Shanghai Normal University)
This presentation examines an adaptive finite element method for the Borrvall-Petersson problem in topology optimization of fluids. In addition to a box constraint and a volume restriction, the model features regularization of the material distribution field by an inverse permeability term so that the optimality system consists of generalized Stokes equations and a variational inequality. The velocity field is approximated by the nonconforming linear element (Crouzeix-Raviart element) while the piecewise constant is applied to the pressure field and the material distribution, respectively. A standard adaptive algorithm is devised for efficient numerical simulation of the model and is proved to generate a null sequence of estimators and a subsequence of discrete solutions converging to some solution of the optimality system. Numerical results illustrate the effectiveness of the proposed algorithm.
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Advances in Numerical Methods for Topology Optimization: Applications Across Disciplines
Xiaoping Wang (Chinese University of Hong Kong (Shenzhen))
This talk explores the cutting-edge numerical methods and their diverse applications in the field of topology optimization. We will highlight the development of robust numerical strategies designed to tackle the challenges that arise from the high dimensionality and non-linearity inherent in topology optimization problems. The iterative thresholding method will be discussed for its significant improvements over conventional techniques, enhancing both the efficiency and stability of solutions. Additionally, the presentation will highlight innovative research integrating machine learning with numerical methods to accelerate the process of topology optimization.
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PRNN-STO: A parametric representation-oriented neural network method for Shape and topology optimization
Jizu Huang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
In this talk, we propose a parametric representation-oriented neural network method for shape and topology optimization problems (PRNN-STO), which consists of the following key components. First, this method uses parametric representations combined with logical operations to effectively capture topological changes within the design domains, thereby reducing the dimensionality of the optimization space and enhancing its ability to represent complex topologies. Second, the Karush--Kuhn--Tucker (KKT) system arising from the optimization problem is solved by integrating the Alternating Direction Method of Multipliers (ADMM) and the Physics-Informed Neural Network (PINN). Due to its mesh-free nature, the proposed method avoids mesh-related issues inherent to traditional FEM-based approaches. Numerical examples demonstrate the accuracy, robustness, and effectiveness of PRNN-STO across a range of shape and topology optimization problems.
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A penalty method for topology optimization problems
Wei Gong (Chinese Academy of Sciences)
We propose a novel penalty framework for compliant mechanism problems, incorporating a convex nonlocal perimeter approximation. We rigorously analyze the existence of solutions to the optimization problem derived from the penalty method. Furthermore, we establish that the discrete problem Gamma-converges to the continuous problem. To solve the discrete problem, we develop a projected gradient method that guarantees strict monotonic descent of the objective function. Numerical experiments on the compliant mechanism and heat dissipation problems validate the effectiveness of the proposed method. We also generalize this approach to minimum compliance problems with pointwise stress constraints. This is a joint work with Yuanda Ye.
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Phase-Field Topology Optimization for Thermo–Fluid Coupled Systems
Shengfeng Zhu (East China Normal University)
We present a phase-field based framework for topology optimization in thermo–fluid coupling systems. The existence of optimal solutions is established, and the Fréchet differentiability of the state variables with respect to the design field is proven. Based on this, we derive gradient flow formulations for both smooth and obstacle-type potentials. A stabilized semi-implicit scheme is introduced, while a primal–dual active set strategy is employed to handle variational inequalities. Numerical results demonstrate the efficiency and robustness of the proposed approach.
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Topological and Shape Optimization Methods for Inverse Shape Reconstruction
Julius Fergy Rabago (Kanazawa University)
This talk focuses on inverse reconstruction problems governed by elliptic partial differential equations, where the objective is to identify unknown subregions associated with discontinuous coefficients from boundary measurements. We discuss numerical methods based on topological sensitivity analysis and shape optimization for recovering the location and geometry of hidden structures. Particular emphasis is placed on the development of efficient reconstruction algorithms, robustness with respect to noisy measurements, and applications to interface detection problems arising in applied sciences and engineering. This is based on joint work with L.~Afraites (Université Sultan Moulay Slimane, Morocco), A.~Hadri (Université Ibn Zohr, Morocco), and M.~Hrizi (Université de Monastir, Tunisia).
MS17 Verified and High-Performance Numerical Computation: Nonlinear Problems, Iterative Methods, and Matrix Multiplication Emulation Main organizer: Katsuhisa Ozaki (Shibaura Institute of Technology) · 4 talks
Organizers
Main organizer: Katsuhisa Ozaki (Shibaura Institute of Technology)
Program & Speakers
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Matrix Factorizations Using Preconditioning and Ozaki Schemes
Katsuhisa Ozaki (Shibaura Institute of Technology)
This talk discusses Ozaki schemes for high-precision matrix multiplication and their applications to numerical linear algebra on NVIDIA GPUs. Motivated by the widening performance gap between low-precision Tensor Core operations and native FP64 arithmetic, we show how emulated matrix multiplication can exploit fast INT8 Tensor Cores to obtain FP64-like accuracy. We introduce the idea of repeatedly using high-speed GEMM engines and compare two approaches: the slice-based Ozaki-I scheme and the CRT-based Ozaki-II scheme. Benchmark results demonstrate the potential performance of these schemes for large DGEMM problems. Finally, we discuss preconditioning strategies for Cholesky, LU, and QR factorizations, where preconditioners are used to avoid inefficient products of tall-and-skinny and short-and-wide matrices and to expose larger GEMM-dominant computations.
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A Library for Emulating BLAS-like Matrix Operations Using the Ozaki-II Scheme
Yuki Uchino (RIKEN Center for Computational Science)
Modern computing architectures feature low-precision matrix multiplication units that achieve substantially higher throughput than their high-precision counterparts. Motivated by this architectural trend, we proposed the Ozaki-II scheme, a high-throughput emulation method for high-precision matrix multiplication using low-precision matrix multiplication units. In this talk, we introduce GEMMul8, a library that provides emulation of BLAS-like matrix operations based on the Ozaki-II scheme. We also present performance evaluations on modern GPUs of the proposed library.
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Shifted Bi-CG method combined with the cross-interactive residual smoothing
Kensuke Aihara (Tokyo City University)
We consider solving shifted linear systems with a large sparse nonsymmetric matrix using Lanczos-type iterative methods. Cross-interactive residual smoothing (CIRS) is an effective technique for obtaining a smooth convergence behavior of residual norms, and results in a smaller residual gap (the difference between the recursively updated residual and the explicitly computed residual), when solving the standard single linear systems. In this talk, we propose applying CIRS to the shifted bi-conjugate gradient (Bi-CG) method. Several implementations of combining CIRS with the seed and shifted parts of algorithms are presented, and their computational efficiency and the residual gap are evaluated through numerical experiments.
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An improved scheme for rigorous integration of semilinear parabolic PDEs
Akitoshi Takayasu (University of Tsukuba)
In this talk, we present a rigorous integrator for the study of initial and boundary value problems for semilinear parabolic PDEs. Our method is based on a non-autonomous semigroup approach, in which the solution operator of the linearized PDE along a numerically computed time-dependent approximate solution is rigorously controlled. In particular, we introduce a multi-stepping scheme based on a new construction of the multi-step propagator via direct backward iteration, allowing rigorous integration over much longer time intervals. For each time step, the dynamics are decomposed into a signed finite-dimensional bare propagator and a finite-tail coupling correction. The bare propagators are composed at the signed matrix level, preserving cancellation effects and significantly reducing the wrapping effect. As a result, the proposed method enables long-time rigorous integration of the Ohta–Kawasaki equation. This is joint work with Jean-Philippe Lessard (McGill University).
MS18 Recent Advances in Computational Science with Machine Learning Main organizer: Seungchan Ko (Inha University) · 8 talks
Organizers
Main organizer: Seungchan Ko (Inha University)
- Co-organizer: Changhoon Song (Research Institute of Mathematics, Seoul National University)
Program & Speakers
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Learning interpretable dynamics
Qianxiao Li (National University of Singapore)
We discuss some recent work on constructing stable and interpretable macroscopic and mesoscopic dynamics from trajectory data using deep learning. We adopt a hypothesis-driven approach: contrary to using generic neural networks as functional approximators for the vector fields driving a dynamical system or operator based learning approaches, we start with a structured hypothesis class in which well-posedness and numerical stability are a priori ensured. Learning on trajectory data then identifies a specific member of this family suitable for capturing the observed dynamical process, from which scientific predictions and analysis can be made. We demonstrate this approach on a varieity of macroscopic and mesoscopic modelling problems, including polymer chain dynamics, dynamics of magnetic systems and self-healing in high-entropy alloys.
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Toward Foundation Models for AI Weather Prediction: Extreme Rainfall, Super-Resolution, and Global Ensemble Forecasting
Youngjoon Hong (Seoul National University)
Artificial intelligence is rapidly transforming weather prediction, from regional nowcasting to global ensemble forecasting. In this talk, I will present our recent efforts toward generative weather foundation models across multiple scales. I will first introduce exPreCast, an AI-based model for precipitation nowcasting and extreme rainfall prediction. I will then discuss a generative super-resolution framework that converts low-resolution deterministic forecasts into high-resolution probabilistic predictions, aiming to recover fine-scale rainfall structures and represent uncertainty in extreme events. Finally, I will present our ongoing ERA5-based global medium-range ensemble forecasting model. The talk will highlight key challenges, including spatiotemporal dynamics, uncertainty quantification, physical consistency, and reliable generalization.
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Learning Stiff Plasma Global Models with PS-PINNs and DeepONet Surrogates
Heejae Kwon (Heejae Kwon)
Plasma global models describe low-temperature plasma dynamics as stiff, multi-species ODE systems. Although numerical solvers are reliable, parameter studies require repeated simulations across operating conditions, motivating SciML approaches for efficient solution and surrogate modeling. This talk presents two approaches. First, PS-PINNs improve PINN training for stiff chlorine discharge dynamics through input preprocessing and adaptive residual balancing, and are demonstrated on forward, inverse, and parameter-dependent problems. Second, a DeepONet surrogate learns parameter-to-trajectory maps for gas-specific global models from solver-generated data using time-dependent normalization and a staged hybrid loss. In silane and chlorine examples, the proposed models reproduce transient plasma state trajectories over operating-condition spaces, suggesting SciML as a reusable framework for plasma chemistry simulation.
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A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networks
Myeong-Su Lee (Seoul National University)
While Physics-Informed Neural Networks offer a promising framework for partial differential equations, the standard L2 residual loss is insufficient for solving the Bhatnagar-Gross-Krook model of the Boltzmann equation. Specifically, small L2 residuals do not guarantee small errors in macroscopic moments, causing the standard PINN to fail in capturing the true solution. To resolve this, we propose a weighted L2 loss function based on a stability analysis. We obtain that minimizing this weighted loss strictly controls the distance between the exact and approximate solutions. Numerical experiments also show that employing this theoretical guided PINN loss leads to superior accuracy across various benchmarks compared to the standard PINN approach.
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Robust training of physics-informed neural networks for p-Laplace problem with rigorous error estimates
Kyueon Choi (Yonsei University)
While physics-informed neural networks (PINNs) are widely adopted, most studies remain confined to linear operators, and a substantial gap persists with classical numerical analysis due to the lack of theoretical error estimation. To bridge these gaps, we propose a robust training framework and establish rigorous error estimates for PINNs solving nonlinear PDEs. In particular, we introduce a dual-norm residual estimation for the p-Laplace equation to overcome challenges posed by low regularity. More precisely, while standard PINNs require strong solutions satisfying the PDE almost everywhere, our framework accommodates cases where solutions exist only in a weak sense. Furthermore, we employ fractional Sobolev norms on the boundary to achieve rigorous estimates. Specifically, within this framework, we derive both a priori and a posteriori error bounds. In addition, the analysis is generalized to a parametric framework. Finally, numerical experiments validate our theoretical estimates.
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Scientific AI for High-Dimensional PDE Learning and Equation Discovery
Jae Yong Lee (Chung-Ang University)
Recent advances in artificial intelligence have opened new opportunities in scientific computing and scientific discovery. In this talk, I present two recent directions in Scientific AI. First, I discuss a BSDE-based learning framework for solving high-dimensional partial differential equations (PDEs), including an unbiased and second-order-free training method that improves computational efficiency while maintaining accuracy. Next, I introduce DoLQ, a framework for discovering ordinary differential equations (ODEs) from observational data by combining symbolic regression with large language model (LLM)-based qualitative reasoning. Together, these works demonstrate how modern AI can contribute to both scalable scientific computing and scientific discovery.
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Why Agentic Theorem Prover Works: A Statistical Provability Theory of Mathematical Reasoning Models
Sho Sonoda (RIKEN AIP)
We develop a statistical theory of LLM-guided theorem proving with proof assistants such as Lean and Rocq. We formulate interactive theorem proving as a finite-horizon deterministic Markov Decision Process, and a tactic-proposing LLM as a stochastic policy with trainable parameters. Statistical provability is defined as the average probability of reaching a proof goal over a distribution of problems. Although theorem proving is hard in the worst case, modern AI-assisted provers often succeed in practice. We explain this gap by studying biased problem distributions. We formalize two types of bias and show that, when the distribution is sufficiently biased, learning can make statistical provability positive: the model can prove new theorems similar to those it has seen. For hierarchical problems, hierarchical LLMs can improve learning efficiency, giving a principled justification for subgoal decomposition in agentic theorem provers. Joint work with Shunta Akiyama and Yuya Uezato.
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Combining classical numerical schemes with neural networks for PDEs
Jinsil Lee (Seoul National University)
This study analyzes the distinct numerical characteristics of standard Partial Differential Equations (PDEs) and Singularly Perturbed PDEs (SPPDEs) and proposes a hybrid numerical-neural framework for their integrated resolution.While conventional numerical methods exhibit high performance in standard PDE analysis, they encounter chronic issues—such as numerical oscillations and solution distortions—in SPPDEs under extreme conditions where the perturbation parameter (ϵ ) is very small. These failures arise from the inability to accurately capture sharp gradients within the boundary layer. To overcome these limitations, this research presents a hybrid strategy that synergizes classical numerical techniques with artificial neural networks. Furthermore, a rigorous convergence analysis was conducted to establish the mathematical validity of the proposed methodology. The analysis confirms that the hybrid approach ensures stability.
MS19 Recent Advances in Matrix Factorization: Theory and Algorithms Main organizer: Jie Meng (Ocean University of China) · 6 talks
Organizers
Main organizer: Jie Meng (Ocean University of China)
- Co-organizer: Zhigang Jia (School of Mathematics and Statistics & RIMS, Jiangsu Normal University)
- Co-organizer: Hyun-Min Kim (Department of Mathematics, Pusan National University)
Program & Speakers
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Linear-map-based tensor CUR decomposition for robust video foreground-background separation
Susana López-Moreno (Pusan National University)
The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends the notion of matrix multiplication to higher order tensors, and which is a generalization of the T-product. Under this framework, we introduce the tensor CUR decomposition, show its performance in video foreground-background separation for different linear maps and compare it to a robust matrix CUR decomposition, a tensor CUR decomposition, other tensor approximations and the slice-based singular value decomposition. We also provide a theoretical analysis of our tensor CUR decomposition, extending classical matrix results to establish exactness conditions and perturbation bounds.
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Component-wise accurate computation of the square root of an M-matrix
Jie Meng (Ocean University of China)
Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P,\bsu,\bv)$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and $\bsu>0$, $\bv\ge 0$ are two vectors such that $A\bsu=\bv$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. Structure-preserving doubling algorithm and fixed-point iterations are also provided. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.
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Numerical methods approach to discovering low-rank tensor decompositions
Taehyeong Kim (Kyungpook National University)
Exact low-CP-rank decompositions of structured tensors are central problems in algebraic complexity. Numerical decompositions are useful in this setting only when they can be converted into exact symbolic identities. We introduce several numerical approach framework based on sparse optimization and Sigma-Pi-Sigma neural networks for this task. Notably, in the determinant case, we prove $\operatorname{rank}(\det_4) \leq 12$ and $\operatorname{rank}(\det_5) \leq 42$ by finding their exact new formula over fields of characteristic not equal to 2.
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A Novel Adaptive Low-Rank Matrix Approximation Method for Image Compression and Reconstruction
Zhigang Jia (Jiangsu Normal University)
Low-rank matrix approximation plays an important role in various applications such as image processing, signal processing and data analysis. The existing methods require a guess of the ranks of matrices that represent images or involve additional costs to determine the ranks. A novel efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) is proposed to compute the optimal low-rank matrix approximation with adaptive identification of the optimal rank. By introducing a randomized basis extraction mechanism, EOD-ABE eliminates the need for additional rank determination steps and can compute a rank-revealing approximation to a low-rank matrix. Experimental results demonstrate the superior speed, accuracy and robustness of EOD-ABE and indicate that EOD-ABE is a powerful tool for fast image compression and reconstruction and hyperspectral image dimensionality reduction in large-scale applications.
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Generalized singular value decompositions of dual quaternion matrices and beyond
Ling Si-Tao (China University of Mining and Technology)
In high-dimensional data processing and data analysis related to dual quaternion statistics, generalized singular value decomposition (GSVD) of two or more dual quaternion matrices is an essential numerical linear algebra tool for an elegant problem formulation and numerical implementation. In this talk, building upon the existing singular value decomposition (SVD) of a dual quaternion matrix, we put forward several types of GSVD of dual quaternion data matrices in accordance with their dimensions. Due to the peculiarity of containing infinitesimal part for dual quaternion matrices, the obtained series of GSVD of dual quaternion matrices dramatically distinguish from those in the real field, the complex field, and even the quaternion ring, but can be treated as an extension of them in some sense.
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On Two-Stage Householder Orthogonalization
Meiyue Shao (Fudan University)
Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix $A$ against another matrix $V$ with orthonormal columns. A common approach is to employ the block Gram–Schmidt algorithm. However, its stability largely depends on the condition number of $[V,A]$. While performing a Householder orthogonalization on $[V,A]$ is unconditionally stable, it does not utilize the knowledge that $V$ has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire $V$, our algorithm only needs to orthogonalize a square submatrix of $V$. Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.
MS20 Topological data analysis and machine learning Main organizer: Jae-Hun Jung (POSTECH) · 7 talks
Organizers
Main organizer: Jae-Hun Jung (POSTECH)
- Co-organizer: Shizuo Kaji (Kyoto University)
- Co-organizer: Kelin Xia (NTU)
Program & Speakers
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Persistent Homology with Path-Representable Distances on Graph Data
Byeongchan Choi (KIAS)
Persistent homology (PH) is widely applied to graph data, but how the choice of graph distance affects the resulting barcodes has received little attention. We define path-representable distances on graphs and compare their persistent homologies. We identify a nontrivial injection between the 1-dimensional barcodes of the unweighted and weighted shortest-path distances, give sufficient conditions via a subclass, cost-dominated distances, and show it holds in dimensions 0 and 1 but not higher. To quantify these correspondences, we introduce the total persistence difference (TPD), a new measure of the changes between filtrations induced by cost-dominated distances on a fixed graph. We prove a stability result for TPD when the distances admit a partial order, and apply it to the SNAP EU research-institution email dataset. TPD captures both periodic patterns and global trends, and aligns more strongly with classical graph statistics than an existing PH-based measure on the same data.
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Periodic Topological Deep Learning for Polymer Design and Discovery
Yasharth Yadav (NTU)
Polymers are central to energy, healthcare, and materials science, but their vast chemical space makes discovery challenging. Most machine learning models represent polymers as a monomer graph, missing chain periodicity and many-body interactions. I will introduce Periodic-TDL, a deep learning framework using periodic Rips complexes and hierarchical simplicial message passing to capture interactions across spatial scales. Periodic-TDL achieves state-of-the-art performance across electronic, optical, physical, and thermal property prediction tasks. We then use it to test two design principles for thermal stability, ester-to-amide substitution and backbone alpha-methylation. Across 48,208 generated polymers, the model captures systematic changes in glass transition temperature arising from these functional group modifications. Experiments on six novel polymer pairs, including three newly synthesized polymers, confirm the predicted trends, supporting topology-informed polymer design.
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Learning Long-Range Connectivity with Neural Networks
Meiyan Kang (Kyoto University)
Deep neural networks (DNNs) are highly effective at learning local image features such as texture, yet some image analysis tasks require an understanding of global connectivity. Such information cannot be inferred from local cues alone and is therefore challenging for standard neural networks. In this talk, we propose an auxiliary, or pretext, task for learning connectivity. Given an image and a pair of points, the network predicts whether the two points belong to the same connected component. We also construct a synthetic benchmark dataset whose intricate patterns require long-range connectivity reasoning. Since connectivity is mathematically computable, labels can be generated not only for synthetic images but also for images from arbitrary domains. This makes the method a form of task augmentation rather than data augmentation. We evaluate its effectiveness on several downstream tasks. The results show that connectivity-aware training improves performance.
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Provably Correct k-Means Clustering of Persistence Diagrams
Hajin Lee (Korea University)
We study k-means clustering of persistence diagrams through their Hilbert-space embeddings, focusing primarily on persistence landscapes. Although persistence landscapes are widely used in practice, it remains unclear when clustering them faithfully reflects clustering in the diagram space. We give simple, verifiable geometric conditions under which (i) nearest-center labels in the landscape space exactly agree with those in the diagram space, and (ii) the landscape k-means objective provably calibrates the diagram-space objective, leveraging tools from modern statistical learning theory. Combined with known fast rates for k-means in Hilbert spaces, our results show that landscape-based k-means provides a statistically and computationally efficient surrogate for diagram-space k-means with explicit performance guarantees and a controllable geometric distortion.
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Topological Regularisation in Basis Learning
Shizuo Kaji (Kyoto University)
We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Many data modalities can be viewed as non-negative functions on structured domains, where basis quality is intrinsically linked to its topology. However, standard methods for incorporating support topology are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier, designing topological scores that integrate into the NMF objective as regularisers. The resulting framework unifies the modelling of spatially coherent image components, periodic time-series structures, and clique-like graph signals.
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Topological Data Analysis for Feature Extraction in Machine Learning
Jisu Kim (Seoul National University)
Topological Data Analysis (TDA) offers a powerful framework for capturing the shape of data. Its core method, persistent homology, observes data at multiple scales to summarize topological features—such as clusters, loops, and voids—that persistently appear. While these summaries provide noise-robust, coordinate-free insights, integrating such non-Euclidean structures directly into machine learning remains challenging. This talk explores using TDA as a robust feature extraction tool to bridge this gap. I will examine featurization techniques that map persistence-based information into Euclidean or functional spaces for seamless integration into standard learning algorithms, particularly deep learning. Specifically, I discuss transforming topological summaries into differentiable neural network layers, alongside geometric representations for visualization. Through these methods, I demonstrate how topological features enhance model representation power and interpretability.
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Persistent Cross Entropy for Causal Inference in Dynamical System
Jae-Hun Jung (POSTECH)
Inferring causal relationships between dynamical system time-series is important in many applications. Topological approaches assume that observed data are measurements of underlying manifolds and that the topology of reconstructed embedding spaces contains information about causal structure. We propose a novel causal inference method based on persistent cross entropy(PCE). We introduce a mathematically consistent probabilistic formulation for comparing persistence diagrams, which remains well-defined even for diagrams of different sizes. This enables principled comparison of topological information from different dynamical systems. Based on this, we develop a topology-based method for causal inference between dynamical system signals. Numerical experiments show that the proposed method captures causality more effectively than methods based on bottleneck and Wasserstein distances, persistence images, and persistence silhouettes.
MS21 Recent developments in sampling methods, uncertainty quantification, and generative AI Main organizer: Zhiwen Zhang (The University of Hong Kong) · 7 talks
Organizers
Main organizer: Zhiwen Zhang (The University of Hong Kong)
- Co-organizer: Liu Liu (The Chinese University of Hong Kong)
- Co-organizer: Zhongjian Wang (Nanyang Technological University)
Program & Speakers
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Asymptotic-Preserving Hybrid UQ Methods for Uncertain Multi-Scale Kinetic Equations
YIWEN LIN (Shanghai Jiao Tong University)
This work develops efficient uncertainty quantification methods for multi-scale kinetic equations with random inputs. For the uncertain Boltzmann equation, we propose hybrid-based multi-level Monte Carlo and bi-fidelity approaches that integrate asymptotic-preserving schemes with hydrodynamic limit solvers. For the semiclassical Schrödinger equation, we design multi-fidelity methods adopting grid-based, meshless and level set solvers for its semiclassical limit. We also investigate the multi-phase Navier-Stokes-Vlasov- Fokker-Planck system with random initial inputs, which serves as an application of multi-scale physical problems. Numerical experiments demonstrate that the proposed methods achieve significant computational speedups over standard methods without sacrificing numerical accuracy. Preliminary inverse results for kinetic equations are also presented.
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A multiscale reduced method for the semiclassical Schr\"odinger problems with random potentials
Panchi Li (Soochow University)
I will present a multiscale reduced-order method for solving the semiclassical Schrödinger equation with random potentials. The proposed approach combines multiscale finite element methods (MsFEM), proper orthogonal decomposition (POD) and the quasi-Monte Carlo (qMC) sampling method. It can be applied to both linear eigenvalue problems and nonlinear time-dependent problems. The total approximation error is controlled by the truncation dimension, the coarse mesh size, the POD error and the qMC sample size. Numerical experiments demonstrate that the method achieves high accuracy while significantly reducing the computational cost compared to standard finite element methods. Additionally, a time-splitting scheme that preserves convergence for discontinuous potentials is introduced for the nonlinear Schrödinger equation. This framework provides an efficient and reliable means of simulating semiclassical quantum systems exhibiting high-dimensional randomness and multiscale features.
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Stable derivative free Gaussian mixture variational inference for Bayesian inverse problems
Daniel Huang (Peking University)
Black-box variational inference (BBVI) with Gaussian mixture families offers a flexible approach for approximating complex posterior distributions without requiring gradients of the target density. However, standard numerical optimization methods often suffer from instability and inefficiency. We develop a stable and efficient framework that combines three key components: (1) affine-invariant preconditioning via natural gradient formulations, (2) an exponential integrator that unconditionally preserves the positive definiteness of covariance matrices, and (3) adaptive time stepping to ensure stability and to accommodate distinct warm-up and convergence phases. Numerical experiments on multimodal distributions, Neal's multiscale funnel, and a PDE-based Bayesian inverse problem for Darcy flow demonstrate the effectiveness of the proposed method.
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Classification and generation in open space communication with structured light
Guillaume Bal (University of Chicago)
We study the classification of structured-light beams after propagation through a random turbulent medium, with the turbulence-degraded speckle patterns generated by numerical simulation of a stochastic paraxial wave model. The classification task is formulated over a finite alphabet, and two input representations — raw intensity and autocorrelation — are benchmarked using two CNN architectures, SimpleCNN and ResNet-18, with the latter offering a better trade-off between computational cost and classification accuracy. We also present a physics-aware generative augmentation framework based on denoising diffusion probabilistic models, designed to supplement scarce training sets by generating additional class-conditioned speckle images. This is joint work with Aokun Wang, Anjali Nair, and Zhongjian Wang.
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Accelerated iterative method for solving the steady-state Boltzmann equation
Yanli Wang (Beijing Computational Science Research Center)
In this work, we propose a modified Newton method equipped with a macroscopic synthetic system (Newton-MS) for the steady-state Boltzmann equation with the quadratic collision operator. In Newton-MS, the modified Newton iteration is utilized as the outer nonlinear solver, while each Newton correction equation is solved by an inner source iteration, where the linearized collision operator is utilized to approximate the quadratic collision model, and it is reduced into a linear iteration. Moreover, a macroscopic synthetic system based on Chapman-Enskog closure is derived to accelerate the convergence of the linear inner iteration in the continuum limit. Several numerical examples, including the 1D Fourier, Couette flow problem, and the 2D cavity flow and thermal-driven cavity flow, are studied to validate the high efficiency of Newton-MS.
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Multi-fidelity numerical methods for a class of kinetic models
LIU LIU (The Chinese University of Hong Kong)
In this talk, we will discuss about a class of multi-fidelity numerical methods for solving kinetic models. In the first part, we will address the uncertainty quantification for kinetic problems and development of bi-fidelity or tri-fidelity methods for different models, where some error estimates are studied using the hypocoercivity. In the second part, we will study an efficient asymptotic-preserving scheme for solving the Boltzmann equation with bi-fidelity algorithm designed in the velocity discretization. Some applications to deep learning approaches for kinetic models will also be discussed, with the idea of bi-fidelity introduced. These are joint works with Xueyu Zhu, Lorenzo Pareschi, Nicolas Crouseilles, Zhen Hao and Zhenyi Zhu.
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Posterior learning by Mean Flow
Zhongjian Wang (Nanyang Technological University)
In this talk I will explain how to use the diffusion model, especially mean flow, to construct unbiased, efficient Bayesian posterior sampler. If time permits, I will also talk a bit on the regularity of the mean flow maps.
MS22 Learning-Based Scientific Computation: Theory, Algorithms, and Applications Main organizer: Aiqing Zhu (Department of Mathematics, National University of Singapore) · 4 talks
Organizers
Main organizer: Aiqing Zhu (Department of Mathematics, National University of Singapore)
- Co-organizer: Qianxiao Li (Department of Mathematics, National University of Singapore)
- Co-organizer: Yue Zhao (Institute for Functional Intelligent Materials, National University of Singapore)
Program & Speakers
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Rhythm-ONet: A query-conditioned neural operator for high-fidelity emulation of multiscale rhythm transitions in neurological disorder modeling
silu Yan (Great Bay University)
Mechanistic neural mass models (NMMs) are essential for digital twins in neurological disorders, but their dynamics can vary sharply with synaptic connectivity, making parameter sweeps costly and challenging for standard neural operators due to spectral bias. We propose Rhythm-ONet, a query-conditioned neural operator with a time-conditioned coefficient generator in the branch pathway. Inspired by attention, it dynamically reweights branch features according to query points, while a Fourier-embedded trunk improves reconstruction of multiscale rhythms. Tested on the Jansen–Rit model for Alzheimer’s-like activity and the Wendling–Chauvel model for seizure-like dynamics, Rhythm-ONet achieves lower relative L2 error than DeepONet, Fourier-DeepONet, and DeepONet-Bt, with better time-frequency agreement across connectivity regimes. Once trained, it enables rapid mapping of connectivity-dependent rhythm transitions, including rhythm slowing in AD and ictal-like discharges.
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Learning and Data Assimilation McKean-Vlasov Dynamics from Particle Data
Liyao Lyu (University of Science and Technology of China)
Learning interaction laws in collective systems and incorporating partial observations into mean-field dynamics are two closely related challenges. We propose a unified framework that addresses both. First, we introduce a measure-valued neural network (MVNN) that learns measure-dependent interaction (drift) terms directly from particle trajectories, using scalable distribution-to-vector embeddings. We establish well-posedness, propagation of chaos, and universal approximation with quantitative rates under a low-dimensional structure. Second, we develop Multiscale Nudge, a data assimilation method that injects macroscopic observations into microscopic dynamics via a Wasserstein gradient-based nudging term with a tractable particle implementation. Under suitable assumptions, we prove exponential decay of the $L^2$ error up to a model-misspecification bias.
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Simulation-Free Dynamic Unbalanced Optimal Transport
Peijie Zhou (Peking University)
The Wasserstein–Fisher–Rao (WFR) metric extends dynamic optimal transport to systems with both displacement and mass variation, but existing solvers are often unstable and costly. We introduce WFR Flow Matching (WFR-FM), a simulation-free method that learns a transport vector field together with a scalar growth rate for birth–death dynamics. We prove that minimizing the WFR-FM objective recovers WFR geodesics. In single-cell applications, WFR-FM reconstructs trajectories involving proliferation and apoptosis, estimates time-dependent growth fields, and supports generative modeling under imbalanced data. Across benchmarks, it improves efficiency, stability, and reconstruction accuracy over existing methods. WFR-FM thus provides a scalable framework for learning continuous dynamics from unbalanced snapshots in which both states and mass evolve.
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Hypothesis-driven construction of mesoscopic dynamics
Zhuoyuan Li (National University of Singapore)
Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. We empirically validate that the proposed approach acts as an effective dynamics learner and offers vital interpretable diagnostics of the underlying physics.
MS23 Recent Advances in Modeling and Numerical Methods for Interface Problems Main organizer: Yu-Hau Tseng (National University of Kaohsiung, Taiwan) · 4 talks
Organizers
Main organizer: Yu-Hau Tseng (National University of Kaohsiung, Taiwan)
- Co-organizer: Wei-Fan Hu (Department of Mathematics, National Central University, Taiwan)
- Co-organizer: Hui-Lan Chang (Department of Applied Mathematics, National University of Kaohsiung, Taiwan)
Program & Speakers
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Triplet approximation in Schloegl’s second model for non-equilibrium phase transitions in lattice bistable systems
Chi-Jen Wang (National Chung Cheng University)
Schloegl’s second model involves two processes: spontaneous particle annihilation at a rate p, and autocatalytic particle creation at empty sites with a rate dependent on the neighboring sites. Stochastic kinetic Monte Carlo simulations are commonly used to understand the model behavior, revealing a discontinuous transition when p surpasses a critical value p_e. To improve upon mean-field and pair approximations, this model employs a triplet approximation, which preserves essential local correlations and yields a more accurate description of spatially inhomogeneous configurations. Our main focus is on analyzing the dynamics of interfaces using discrete reaction-diffusion equations derived from higher-order triplet approximations to the exact master equations for spatially inhomogeneous states. This study contributes to a better understanding of phase coexistence and the discontinuous transition characteristic of such dynamical systems.
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Discrete Leray projection for solving semi-implicit schemes in fluid-structure interaction problems
Yunchang Seol (Chonnam National University)
In this talk, we present an efficient approach for solving semi-implicit schemes of the motion of elastic interfaces in unsteady Navier-Stokes flows, with a focus on immersed boundary methods. In order to solve its original linear system, we transform the 3-by-3 block matrix to a reduced 2-by-2 block matrix after eliminating the fluid pressure term. Such rank reduction is possible by applying the discrete Leray projection operator in a staggered MAC grid. By virtue of this feature, an efficient algorithm for finding two unknowns is suggested using the Schur complement.
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Phase-field projection method for multi-component fluid flows
Youngjin Hwang (Chonnam National University)
We present a phase-field projection method for the multi-component Navier--Stokes--Cahn--Hilliard system. The Cahn--Hilliard equation is discretized by a convex splitting method, and the Navier--Stokes equations are treated by a fractional step method. Through this decomposition, the coupled system is solved sequentially by several subproblems instead of one fully coupled problem. The proposed method satisfies the discrete incompressibility condition and mass conservation. Various numerical experiments are performed to demonstrate the main idea of the projection method, and the discrete energy dissipation and mass conservation are observed numerically. The method can be further extended to problems with variable mobility and density.
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Motion of three geostrophic vortices on a sphere
Habin Yim (Chonnam National University)
Accurately approximating and applying geophysical phenomena requires considering geostrophic vortices on spherical or sphere-like curved surfaces. Accordingly, this study explores the dynamics of geostrophic fluid vortices on the surface of a sphere. The geostrophic vortex model incorporates the effects of stratification and planetary rotation and is relevant to large-scale atmospheric phenomena. The governing equations of the model are expressed in terms of the Legendre functions and systematically formulated along with the corresponding Hamiltonian. We focus on the three-vortex case and investigate its dynamics, identifying the fixed and relative equilibria of the vortices. The general dynamics is analyzed using the phase diagram based on trilinear coordinates. Notably, we demonstrate the occurrence of the non self-similar collapse of the three geostrophic vortices. Remarkably, we find a recurring motion of vortices converging toward and diverging from a relative equilibrium.
MS24 High Order Numerical Methods: Design, Analysis, and Applications Main organizer: Xinghui Zhong (Zhejiang University) · 8 talks
Organizers
Main organizer: Xinghui Zhong (Zhejiang University)
- Co-organizer: Juan Cheng (Capital Normal University)
- Co-organizer: Jie Du (East China Normal University)
Program & Speakers
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Analysis of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear fourth-order time-dependent problems
Xiong Meng (Harbin Institute of Technology)
In this talk, we study the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The numerical flux of the nonlinear convection term is the generalized local Lax-Friedrichs flux, and the generalized alternating fluxes are adopted for the fourth- and second-order terms. A proper numerical initial condition is designed based on a modified projection of the third-order derivative. By using the generalized Gauss-Radau projections, together with sharp bounds of nonlinear and jump terms, optimal error estimates are derived. The results are extended to equations with an additional dispersion term and mixed boundary conditions. Examples including Dirichlet as well as generalized Dirichlet boundary conditions, singularly perturbed problems and two-dimensional problems are also numerically investigated, indicating that the theoretical results hold for more general cases.
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Optimal error estimate of a discontinuous Galerkin method for one-dimensional linear hyperbolic equation with degenerate points moving along space-time curves
Qiang Zhang (Nanjing University)
In this talk, we consider a discontinuous Galerkin method with purely upwind numerical flux to solve one-dimensional linear variable-coefficient hyperbolic equation, where the flow speed has different positive and negative signs when passing through the degenerate points. Our purpose is to give a rigorous proof of the optimal L$^2$-norm error estimate when the degenerate points move along smooth curves depending on both space and time. The main difficulty is how to deal with the signs' change of the flow speed regarding time. To this end, we propose a novel analysis framework with the help of a time-dependent projection based on the hybrid application of Gauss--Radau projections and a sharp boundedness concerning the accumulation of all involved jumps on troubled locations.
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High-order inverse Lax-Wendroff procedure for compressible fluid-structure interaction problems
Yan Jiang (University of Science and Technology of China)
In this talk, we present a global high-order method for fluid–structure interaction (FSI) involving compressible inviscid flows and deformable elastic solids. A partitioned strategy couples the fluid and solid solvers. The fluid is discretized by a high-order finite difference WENO method on fixed Cartesian Eulerian grids, while the solid is solved using a Lagrangian discontinuous Galerkin method on unstructured meshes. To accurately treat the moving interface, we develop a high-order inverse Lax–Wendroff based interface treatment, eliminating mesh regeneration and sub-iterations at each time step. The proposed interface treatment also provides robust stability for challenging cases, including light solids interacting with heavy fluids, as supported by linear stability analysis. Numerical results demonstrate third-order accuracy for smooth solutions, non-oscillatory shock-capturing capability, and stable performance over a wide range of material parameters.
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An oscillation-free discontinuous Galerkin method for hyperbolic conservation laws
Jianfang Lu (SCUT)
In this talk, we introduce an oscillation-free discontinuous Galerkin (OFDG) method for solving the hyperbolic conservation laws. Typically, the high order linear numerical schemes would generate spurious oscillations in the presence of discontinuous solutions, which can be harmful to the numerical simulation, as they not only generate some non-physical structures but also make the numerical scheme less robust. To overcome this difficulty, we incorporate a numerical damping term into the classic DG formulation to control spurious oscillations. With this modification, the proposed DG method still maintains many good properties, such as the extremely local data structure, conservation, $L^2$-boundedness, optimal error estimates, and superconvergence. Several numerical examples are provided to demonstrate the performance of the scheme and confirm the derived theoretical results, thus validating the effectiveness of the proposed DG scheme.
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Spectral volume methods for hyperbolic equations
Waixiang Cao (EASIAM 2026)
his talk is concerned with the analysis of two spectral volume (SV) methods for hyperbolic equations : one is constructed basing on the Gauss-Legendre points (LSV) and the other is based on the right-Radau points (RRSV). We first prove that for a general nonuniform mesh and any polynomial degree $k$, both the LSV and RRSV methods are stable and can achieve optimal convergence orders in the $L^2$ norm. Secondly, we prove that both methods have some superconvergence properties at some special points. For instances, at the downwind points, the solution of RRSV and LSV methods converges with the order of ${\cal O}(h^{2k+1})$ and ${\cal O}(h^{2k})$, respectively. Moreover, we demonstrate that for constant-coefficient equations, the RRSV method is identical to the upwind discontinuous Galerkin (DG) method. Our theoretical findings are validated with several numerical experiments at the end.
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A penalty based conservative interface treating method based on DG framework for multi-medium problems
Chunwu Wang (University of Aeronautics and Astronautics)
In this work, a conservative sharp interface treating method is proposed for solving compressible multi-medium flow based on based on DG method for multi-medium flow. To overcome the small cell problem, we introduce the penalty into DG method used in the cells near the interface rather than the complicated cell reconstruction. It is found that this method is stable and conservative. Numerical experiments indicate that the proposed method can capture the shock wave and material interface accurately and sharply even for problems with significant density and pressure gradients.
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A Lax-Wendroff type theorem of doubly conservative schemes for degenerate convection-diffusion equations
Juan Cheng (Juan Cheng, Professor, Capital Normal University)
It is well known that a conservative numerical scheme may still fail to converge to a weak solution for degenerate diffusion or convection–diffusion equations. We introduce doubly-conservative (DoC) schemes and establish a Lax–Wendroff-type theorem: any convergent DoC scheme converges to the weak solution. We further develop two types of new high order numerical schemes that remain stable and convergent under diffusion degeneracy. Numerical experiments confirm its optimal convergence rates and validate the theoretical results.
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Efficient and energy stable DLRA method for multiscale kinetic transport equations
Tao Xiong (University of Science and Technology of China)
In this work, we develop an asymptotic preserving dynamic low rank method for multiscale kinetic transport equations. The proposed scheme is unconditionally stable in the diffusive regime, and significantly reduce computational cost via dynamic low rank approximation. Energy stability is established. Numerical experiments confirm the effectiveness and efficiency of the proposed approach.
MS25 Numerical Methods and Computational Modeling for Complex Physical Systems Main organizer: Shuo Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) · 8 talks
Organizers
Main organizer: Shuo Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Co-organizer: Xia Ji (Beijing Institute of Technology)
- Co-organizer: Hehu Xie (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Program & Speakers
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Fast Jacobi Spherical Harmonic Spectral Methods for the FENE Fokker--Planck Equation with Applications to Closure Modeling
Haijun Yu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
We develop efficient spectral methods for the spatially homogeneous Fokker--Planck equation associated with the finitely extensible nonlinear elastic (FENE) dumbbell model. To address the boundary singularities induced by FENE potential, we construct Jacobi–spherical harmonic spectral methods to incorporate the correct boundary behavior through suitably weighted Jacobi polynomial bases, resulting in sparse linear systems that can be solved efficiently. For time discretization, we employ a semi-implicit BDF2 scheme and establish its energy stability. Numerical experiments demonstrate the accuracy and efficiency of the proposed solvers. Using the high-fidelity solutions as benchmarks, we further assess several commonly used closure approximations, including the Peterlin model (FENE-P) and the quasi-equilibrium approximation (FENE-QE). We also propose a neural-network-based implementation of the FENE-QE closure that significantly reduces computational cost while maintaining high accuracy.
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Neural network methods for fractional Laplacian: from quadrature-enhanced MC fPINN to fTNN
Xiaobo Yin (Central China Normal University)
This talk presents two methods on the neural networks for fractional Laplacian. They all adopt a spatially varying radius, yielding a geometry-adaptive three-part decomposition of the fractional Laplacian: singular near-field, regular interior far-field, and analytical exterior far-field. Then, for the first method called quadrature-enhanced Monte Carlo fPINN, Gauss-Jacobi quadrature is used for the singular near-field radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables, combined with a feature-enhanced PINN trial space, achieving 1-3 orders of magnitude improvement over existing baselines on problems up to 10000 dimensions. The second method is a fully deterministic method, called fTNN, which replaces the angular Monte Carlo sampling by deterministic directional quadrature, reaching relative errors on the order of 10^{-4} to 10^{-6} in 1D/2D/3D.
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Orthogonality Preserving Methods for Electronic Structure Calculations
Xiaoying Dai (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
To obtain convergent numerical approximations without orthogonalization operations is of great importance in electronic structure calculations. In this talk, we will introduce an extended gradient flow based Kohn-Sham DFT model, for which we prove that the flow is orthogonality preserving and that the solution evolves to the ground state. With the help of the extended gradient flow based Kohn-Sham DFT model, we propose some iteration schemes for the discretized Kohn-Sham model, which are proved to preserve the orthogonality of the Kohn-Sham orbitals automatically. With our schemes, the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization operations when the initial orbitals are orthogonal.
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Effective Approximations for Hybrid Density Functional Theory Calculations
Fei Xu (Beijing University of Technology)
A novel adaptive finite element algorithm is proposed for hybrid density functional theory to address the high complexity and large dense eigenvalue problems induced by the nonlocal Fock exchange operator. The algorithm adopts the multilevel correction framework to decompose high-dimensional nonlinear eigenvalue problems into linear boundary value problems and low-dimensional subspace problems. A general low-rank approximate exchange operator is constructed, which maintains Hermiticity, guarantees calculation accuracy and avoids matrix singularity. A three-level nested SCF iteration strategy is also introduced to cut the cost of repeated operator construction. This method effectively breaks the computational bottleneck of traditional approaches and offers an efficient numerical solution for electronic structure simulations.
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Quasi-entropy: in free energies and closure approximations
Jie Xu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Molecular-theory-based tensor models of liquid crystals typically contain an explicit entropy term deduced from the maximum entropy state. For dynamic models, high-order tensors also appear in the model, for which a classical closure approximation is also given the maximum entropy state. The maximum entropy state is able to maintain the essential properties and structures of the molecular theory in tensor models, but leads to high computational cost. Quasi-entropy is a class of elementary functions to substitute the terms deduced from the maximum entropy state, which can be incorporated both in free energies and in closure approximations. It not only keeps the desired properties and structures of the models, but also reduces the complexity of dealing with implicit functions involving three dimensional integrals to O(1). We present a few representative results in equilibrium states and dynamics.
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Numerical computation of Bogoliubov collective excitations in Bose-Einstein condensates
Manting Xie (Tianjin University)
Bogoliubov collective excitations represent a distinctive quantum state in Bose-Einstein condensates, which mathematically correspond to a class of linear response eigenvalue problems known as the Bogoliubov-de Gennes equations (BdGEs). This report presents an efficient, robust, and high-precision numerical method for solving such BdGEs. It covers the fundamental underlying mathematical theories and the numerical analysis framework. The superiority of the proposed method in terms of accuracy and computational efficiency is validated via numerical experiments. Numerical results demonstrate that the implemented algorithm can precisely characterize the low-energy excitation spectra, thereby providing reliable numerical support for investigating macroscopic quantum phenomena in ultracold quantum gases.
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Dirichlet Harmonic Solutions via Neumann-Type Curl Potentials: A Unified Natural Deep Ritz Framework on the de Rham Complex
Jiarong Chen (Beijing Institute of Technology)
Deep neural networks show great promise for high-dimensional PDEs, yet enforcing Dirichlet boundary conditions remains a core challenge. This work builds on a key geometric insight: Dirichlet harmonic solutions admit a curl-driven Neumann-type representation, circumventing the case-by-case tuning and convergence imbalance of standard penalty-based methods in high dimensions. Based on the de Rham complex, our unified Natural Deep Ritz Method (NatDRM) for all $d\geq2$ converts Dirichlet constraints into three coupled Neumann subproblems via dimension-adaptive curl potentials, with unified discrete losses, lightweight boundary gauge-fixing regularizations, and extensions to variable-coefficient and semilinear Poisson cases. Numerical experiments up to 6D show the method matches or outperforms optimally tuned DRM and PINN in most cases, with stable convergence and synchronous interior-boundary error decay.
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Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems
Huaijia Zhang (Beijing Institute of Technology)
In this work, we present a machine‑learning method based on tensor neural networks for solving elliptic multiscale problems. Thanks to their special structure, tensor neural network functions allow direct, highly‑accurate high‑dimensional integration without resorting to Monte‑Carlo sampling. Using homogenization techniques, the original multiscale problem is first transformed into a high‑dimensional limit problem with satisfactory accuracy. We then construct a machine‑learning approach based on tensor neural networks to solve this resulting high‑dimensional limit problem. The proposed framework offers a new avenue for developing high‑accuracy numerical methods for more general multiscale problems. Several numerical examples are presented to verify the performance of our numerical scheme.
MS26 Bridging AI and Scientific Computing Main organizer: Cheng Yuan (Wuhan University) · 4 talks
Organizers
Main organizer: Cheng Yuan (Wuhan University)
- Co-organizer: Lican Kang (Wuhan University)
- Co-organizer: Yuling Jiao (Wuhan University)
Program & Speakers
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Standard Transformers Achieve the Minimax Rate in Nonparametric Regression with $C^{s,\lambda}$ Targets
Yanming Lai (The Hong Kong Polytechnic University)
The tremendous success of Transformer models in fields such as large language models and computer vision necessitates a rigorous theoretical investigation. In this work, we show that standard Transformers can approximate Hölder functions $ C^{s,\lambda}\left([0,1]^{d\times n}\right) $$ (s\in\mathbb{N}_{\geq0},0<\lambda\leq1) $ under the $L^t$ distance ($t \in [1, \infty]$) with arbitrary precision. Building upon this approximation result, we demonstrate that standard Transformers achieve the minimax optimal rate in nonparametric regression for Hölder target functions. It is worth mentioning that, by introducing two metrics: the size tuple and the dimension vector, we provide a fine-grained characterization of Transformer structures, which facilitates future research on the generalization and optimization errors of Transformers with different structures. These findings provide theoretical justification for the powerful capabilities of Transformer models.
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Score-Based Sequential Langevin Sampling for Data Assimilation
Zhao Ding (Hong Kong Polytechnic University)
This talk introduces score-based sequential Langevin sampling (SSLS), an approach to nonlinear data assimilation within a recursive Bayesian filtering framework. It decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. Theoretically, we establish convergence guarantees for SSLS in total variation distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method.
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Learning Arbitrary Long‑Range Tails for MLIPs with SOG‑Net
CHEN CHEN (Shanghai Jiao Tong Univeristy)
Long‑range (LR) interactions remain a critical bottleneck for machine‑learning interatomic potentials (MLIPs) due to their non‑local and slowly decaying nature. We present a sum-of-Gaussians neural network (SOG-Net), a lightweight and modular framework that learns general LR tails directly from energy/force data without assuming a predefined decay exponent. The architecture connects local atomic descriptors to multidimensional latent variables and applies trainable sum‑of‑Gaussians multipliers in reciprocal space, evaluated via nonuniform fast Fourier transforms at near‑linear cost. This design adaptively captures arbitrary LR behaviors and is backend‑agnostic, seamlessly integrating with short‑range descriptors such as DP, CACE, and NEP. Extensive benchmarks across diverse systems—from molecular dimers and molten salts to electrolytes and ferroic interfaces—show that NEP‑SOG consistently outperforms both purely local and Ewald‑based models.
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Numerical Methods and Analysis of Computing Quasiperiodic Systems
Shifeng Li (Mathematics and Statistics College of Chengdu University of Technology)
Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys. , 256: 428, 2014], has been proposed to compute quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this report, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.
MS27 Propagation, Patterns, and Spatiotemporal Heterogeneity in Reaction–Diffusion Models for Biology Main organizer: Xiaoqing He (East China Normal University) · 7 talks
Organizers
Main organizer: Xiaoqing He (East China Normal University)
- Co-organizer: Huicong Li (Sun Yat-sen University)
Program & Speakers
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Sharp asymptotics for the KPP equation with some front-like initial data
Mingmin Zhang (University of Science and Technology of China)
In this talk, I will present the first PDE proof for Bramson's $o(1)$ results established in 1983 concerning the sharp asymptotics for the KPP equation under front-like initial data decaying as $x^{k+1}e^{-\lambda_*x}$ and $x^{\nu} e^{-\lambda x}$ as $x$ tends to infinity, with $0<\lambda<\lambda_*=\sqrt{f'(0)}$ and with any fixed $k, \nu\in\mathbb{R}$.
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Effects of temporal variations on wave speeds of bistable traveling waves for Lotka-Volterra competition systems
Weiwei Ding (South China Normal University)
This talk is concerned with the bistable traveling waves for two-species Lotka-Volterra competition systems in time periodic environments. We focus especially on the influence of the temporal period, with existence results established for both small and large periods. We also show the existence of, and derive explicit formulas for, the limiting speeds as the period tends to zero or infinity, and provide estimates for the corresponding rates of convergence. Furthermore, we analyze the sign of wave speed. Based on our explicit formulas for the limiting speeds, we construct an example in which the sign of wave speed changes with the temporal period. This example reveals that temporal variations can significantly influence competition outcomes, enabling different species to become dominant under different periods.
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Global well-posedness and blow-up for a chemotaxis system with indirect signal production in $\mathbb{R}^4$
Rui Tang (Southeast University)
We investigate the chemotaxis system with indirect signal production in $\mathbb{R}^4$: \begin{align*} \left\{\begin{array}{lll} \partial_tu-\Delta u+\nabla \cdot(u \nabla v)=0, \\ -\Delta v=w, \\ -\Delta w=u, \\ u(x,0)=u_0(x). \end{array}\right. \end{align*} This system appears as a special case of an attraction-repulsion chemotaxis system describing the motion of microglia in the central nervous system. Some models of this form are also equivalent to certain reaction-cross-diffusion systems with a mechanism termed distributed delayed diffusion, describing the biased movement of animals due to spatial memory. We prove that for every nonnegative initial datum in $L^1(\mathbb{R}^4)$, this system is globally well-posed if and only if the total mass $\int_{\mathbb{R}^4}u_0(x)\dif x \leqslant (8\pi)^2$. We show upper bound life span estimates for mild solutions of this system with large initial data.
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Critical mass for the Keller-Segel system on closed Riemannian surfaces
Yuxiang Li (Southeast University)
In this talk, we study the parabolic-elliptic Keller-Segel system on closed Riemannian surfaces. It is known that the critical mass for global existence versus finite-time blow-up is $8\pi$ in two-dimensional Euclidean space Blanchet et al (2006) and hyperbolic space Maheux et al (2020). Here, we show a critical mass phenomenon on closed Riemannian surfaces. For any initial mass $M<8\pi$, the solution exists globally and remains bounded by virtue of Moser-Trudinger inequality on compact surfaces established by Yang (2007). On spheres and ellipsoids of revolution, for supercritical mass $M>8\pi$, there exist solutions that blow up in finite time, as proved by a refined weighted moment method. Furthermore, for $x_0$-zonal solutions on spheres and ellipsoids of revolution, blow-up can only occur at $x_0$ or its antipodal point.
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Boundedness and exponential convergence in a quasilinear chemotaxis-haptotaxis system
Tian Xiang (Renmin University of China)
To fill in gaps and provide new insights for qusilinear chemotaxis only system and chemotaxis-haptotaxis system, we study global dynamics in a Neumann initial-boundary value problem (IBVP) for a three component quasi-linear chemotaxis-haptotaxis system. Under more relaxing conditions than previous studies, we obtain global boundedness of classical solutions to the corresponding IBVP, and then, under further smallness conditions which are independent of masses of cells in the strong diffusion case, we show exponential convergence of such bounded solution towards the constant equilibrium. Our boundedness, based on several subtle energy estimates, significantly extend and improve earlier ones, and our convergence of bounded solutions, relying on a new derived nonlinear Poincare type inequality, appears to be new even for quasi-linear chemotaxis-only systems.
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On Keller-Segel Models with Signal-Dependent Motility and External Sources
Jie Jiang (Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences)
In this talk, we report some recent progress on the research of the Keller-Segel model with signal-dependent motility and external sources. By introduction of several auxiliary functions together with applications of comparison techniques and energy methods, we are able to prove in several cases that external sources with super-linear damping term guarantee uniform boundedness of the global classical solution, preventing occurrence of infinite-time blowups in the homogeneous counterpart.
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Spreading dynamics for the Lotka-Volterra system with general initial supports: the strong competition
Hongjun Guo (Tongji University)
This talk concerns the spreading dynamics of a high-dimensional strong competition Lotka-Volterra system where two species initially occupy disjoint measurable (possibly unbounded) subsets in R^N, which are called initial support. Recently, Hamel and Rossi introduced some new geometric notions, such as bounded or unbounded directions and positive-distance interior, for single-species equations with general initial supports. Under these notions and appropriate assumptions, we characterize directional spreading behavior for the two-species system: precise spreading speeds and sets for both species are derived.
MS28 Geometric Shape Generation (I) Main organizer: Yoshiki Jikumaru (Toyo University) · 7 talks
Organizers
Main organizer: Yoshiki Jikumaru (Toyo University)
- Co-organizer: Taiki Yamada (Shimane University)
Program & Speakers
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Discrete curvature analysis of the multiplex network
Taiki Yamada (Shimane University)
Network analysis increasingly requires tools beyond classical centrality, especially for multilayer systems where entities interact through several types of connections. This talk is based on my paper that extends Forman curvature, a combinatorial analogue of Ricci curvature, to multiplex graphs. After introducing a compile-graph framework for doubly-weighted multilayer networks, I defines Forman curvature for both intra-layer and inter-layer edges, and establishes explicit upper and lower bounds. A key invariant, the comprehensive evaluation of a vertex, aggregates inter-layer curvature into a scalar that complements classical centrality by capturing how a node's structural role varies across layers. A three-step algorithm identifies such layer-dependent vertices, validated on Erdős–Rényi compile graphs. Classification experiments further show that curvature-based descriptors achieve high accuracy and combine effectively with graph kernels.
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Optimization Approach for Generating Developable or Nearly Developable Strips Forming Spatially Curved Beams of Architectural Surfaces
Kentaro Hayakawa (College of Industrial Technology, Nihon University)
Geometric and structural design of supporting frames for architectural doubly curved free-form surfaces has been intensively studied, yet balancing geometric complexity and manufacturability remains challenging. While the majority of research and implementations rely on straight or planar curved structural elements, we propose a method for constructing spatially curved beams with developable or nearly developable strips, enhancing both aesthetic quality and structural consistency. Our approach enables the fabrication of spatially curved beams through standard bending processes, such as rolling or press bending, without twisting. An optimization method is introduced to generate developable strips forming beam geometries while preserving near-orthogonality between ruling directions and directrix curves. Continuity of beam surface normals and ruling directions is also ensured at intersections, improving both structural integrity and visual coherence.
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On the governing equations of membrane O surfaces
Yoshiki Jikumaru (Toyo University)
Shapes that utilize curvature to efficiently transmit stress play a crucial role in fields such as shell structures and membrane structures in architectural surface design. Rogers and Schief demonstrated that shapes in which a constant load acts in the normal direction of the membrane, and the principal curvature lines and principal stress lines coincide and are in equilibrium, form an integrable system known as an O surface. A representative example is the constant mean curvature surface, known as a mathematical model of a soap bubble that achieves isotropic and homogeneous stress state for the pressure. In this talk, we show the system of differential equations governing membrane O surfaces and demonstrate that they contain known 'solitonic' equations and constitute a truly 'good' system of equations.
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A discretization principle via permutability
Wayne Rossman (Kobe Univeristy)
Many classes of surfaces have geometric characterizations, including special coordinate properties or curvature properties, that admit a description using integrable systems. The viewpoint of discrete differential geometry often hovers around the definition of integrable-systems-based discrete versions of these surfaces, reflecting the various coordinate and curvature properties. Such discretization has been carried out in a wide variety of cases, including, for example, pseudospherical surfaces, constant mean curvature surfaces, isothermic surfaces, or Ω-surfaces. Here we consider the case of discrete isothermic surfaces arising from the permutability of Darboux transformations of smooth isothermic surfaces so that they are circular nets, and we show the following: permutability of transforms of smooth surfaces with particular characteristics leads to discrete surfaces with discrete analogues of the same characteristics. This is joint work with J. Cho and M. Pember.
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Geometry of aesthetic curves based on self-affinity in Klein geometries
Shun Kumagai (Hachinohe Institute of Technology)
Log-Aesthetic Curve (LAC) is a family of curves proposed by Harada et al. in 1995 as a reference for shapes in industrial design; while offering limited controllability, it was intended as a consistent shape element. The consistency is formulated as invariance under a one-parameter family of affine deformations of its Euclidean Frenet frame, called "self-affinity". Its variation in affine geometry yields a special case of Klein-Lie W-curve that includes the quadratic curve, power function graph, logarithmic spiral, etc. In this talk, we present the result that the LAC meets the "affine W-curve" in the framework of Möbius geometry based on self-affinity, and discuss deformations of them.
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Design and fabrication workflow for kagome gridshells via curvature optimization and curve partitioning
Kazuki Hayashi (Kyoto University)
Kagome geometric patterns are widely used for design, valued for both their aesthetic appeal and structural stability. However, design processes for large-scale Kagome structures that account for surface rigidity, member strength, and fabricability remain underdeveloped. To address this, we present a novel design workflow for Kagome gridshells. First, the vertical equilibrium of a flat triangulated mesh is determined via the force density method, and a Kagome pattern is generated through mid-edge subdivision. Next, the equilibrium configuration is obtained using a surface-constrained B-spline curve optimization algorithm that minimizes discrete Darboux-frame elastic energy through finite-difference gradient descent. Furthermore, we propose an algorithm to partition the grid curves into three groups, considering singularities to avoid internal intersections. For two groups, integrating a scissors mechanism enables flat-plane joint pre-assembly, significantly simplifying fabrication.
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Low-Dimensional Shape Modeling of Robotic Surfaces
Noriyasu Iwamoto (Shinshu University)
Robotic control requires prediction of the robot’s state from actuator inputs and estimation from sensor measurements. Both rely on kinematic models that must represent shape accurately and remain efficient enough for real-time use. This talk focuses on robotic surfaces, thin sheet-like robots whose many spatially distributed degrees of freedom make modeling difficult. Although finite element simulations and boundary-value methods can describe deformation in detail, their computational cost limits their use in control. We therefore explore low-dimensional models that exploit surface geometry to represent shape with few parameters. Two robotic surfaces developed by the speaker are presented: a sheet-like robot composed of interconnected frustum-shaped actuators modeled by conformal mapping, and a boundary-actuated robotic surface using a soap film modeled as a minimal surface. A piecewise constant-mean-curvature model and its applications to shape design and control are also introduced.
MS29 Recent advances on kinetic theory and continuum mechanics Main organizer: Hiroshi Fujiwara (Kyoto University) · 8 talks
Organizers
Main organizer: Hiroshi Fujiwara (Kyoto University)
- Co-organizer: Daisuke Kawagoe (Kyoto University)
Program & Speakers
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Stability of traveling waves on the half-plane without sign condition on the vorticity
In-Jee Jeong (Korea Institute for Advanced Study)
We establish Lyapunov stability of a family of traveling waves without the sign condition on the vorticity, for the two-dimensional incompressible Euler equations on the half-plane. This is done by combining variational principles with Lagrangian bootstrapping arguments which carefully track the location of the negative part of the vorticity. This seems to be the first stability result which does not require the sign condition in the half-plane. This is based on joint works with Ken Abe, Kyudong Choi, Guolin Qin, and Yao Yao.
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Stability and Boundary Interaction of Viscous Shock Waves for the Burgers Equation
Masaki Imagawa (Academia Sinica)
We study the stability and boundary interaction of viscous shock waves for the Burgers equation on the half-line. The initial data are assumed to be close to a viscous shock profile located far from the boundary. We treat both cases where the shock moves away from the boundary and where it moves toward the boundary. In the outgoing case, pointwise estimates for the perturbation of the shock profile are obtained by introducing a suitable shift of the shock location. In the incoming case, the same analysis remains valid only while the shock is sufficiently far from the boundary. After the shock approaches the boundary, we use the Green's function for the linearized equation around a Riemann solution, which is explicitly described via the Hopf--Cole transform. This enables us to capture the transition from a shock layer to a boundary layer. This talk is based on joint work with Shih-Hsien Yu (Academia Sinica, Taiwan).
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The transmission problem with imperfect interfaces of small resistance
Yong-Gwan Ji (Korea Institute for Advanced Study(KIAS))
We consider the transmission problem in presence of interfaces with imperfect bonding. The imperfect bonding condition is characterized by the positive resistance along the interface, which causes discontinuity of the potential across the interface while the flux is continuous. If the interface resistance is zero, then the interface is of perfect bonding, where both the potential and the flux of the solution are continuous across the interface. In this paper, we first construct using layer potentials the solution to the transmission problem with imperfect interfaces. We then prove that the solutions converge in various Sobolev spaces to the solution to the transmission problem with perfect interfaces as the interface resistance tends to zero. In particular, it is shown that the gradient of the solution converges in the uniform norm if the boundary is sufficiently regular. This is joint work with Shota Fukushima and Hyeonbae Kang.
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The global regularity problem for the axisymmetric, swirl-free Euler equation in four and higher dimensions.
Evan Miller (University of Maine)
In this talk, I will discuss some recent progress on the regularity problem for axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. This problem is more singular than the classical three dimensional case, because the transported quantity may be unbounded initially.
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On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach
Chun-Hsiung Hsia (National Taiwan University)
This is joint work with I-Kun Chen and Daisuke Kawagoe. In this talk, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski in 1987 to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished.
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Schlieren tomography of density in transparent media as the inverse source problem
Hiroshi Fujiwara (Kyoto University)
The Schlieren method is a well-established flow visualization technique that captures the deflection of light caused by refractive index variations in transparent media. By detecting such variations, it enables us to visualize density gradients in fluids, and is widely used, for example, to reveal shock-wave structures. Conventional Schlieren imaging generally produces two-dimensional projections from a single viewing direction. Tomographic Schlieren techniques, which acquire images from multiple view angles, enable the reconstruction of the internal structure of fluid fields. In this study, we develop a novel reconstruction method based on the A-analytic theory. Unlike conventional approaches relying on the Radon transform, the reconstruction problem is formulated as an inverse source problem for the transport equation. Numerical simulation results are presented to demonstrate the effectiveness of the proposed method.
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Uniqueness for alpha-SQG in critical Hölder space
Junha Kim (Ajou University)
In this talk we establish uniqueness for the alpha-SQG equation in the Hölder space $C^{alpha}$. This space is critical in the following sense: a $C^{\alpha}$-solution barely admits a log-Lipschitz velocity, and its associated flow map is uniquely determined. On the other hand, the equation is strongly ill-posed in $C^{\alpha}$, and existence of solutions is not guaranteed for all $C^{\alpha}$-data. Assuming a hypothetical $C^{alpha}$-Lagrangian solution, we prove that it is unique within the class $L^{\infty}_tC^{\alpha}_x$.
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Coefficient Reconstruction in Elliptic and Parabolic PDEs Without Adjoint Problems Using Continuous Data Assimilation
Peiran Zhang (Kyoto University)
In this talk we present a method to reconstruct the spatially distributed diffusion coefficient in elliptic and parabolic partial differential equations (PDEs) from discrete interior measurements of the solutions. The proposed method is derived by the minimization of an error functional. By applying the technique of continuous data assimilation (CDA), an approximate gradient of the error functional is derived, which circumvents the computation of adjoint problems to obtain the gradient required by the conventional approach. We then consider the error analysis of the method using the proposed approximate gradient in a semi-discrete setting. Since the proposed gradient formula is explicit, we are able to investigate directly the descent of reconstruction error, and a Hölder stability is obtained. Finally, some numerical results are also presented, exhibiting the accuracy and stability of the proposed method.
MS30 Non-linear Wave Phenomena: From Quantum Interactions to Stochastic and Transonic Flows Main organizer: Ying-Chieh Lin (Department of Applied Mathematics, National University of Kaohsiung) · 5 talks
Organizers
Main organizer: Ying-Chieh Lin (Department of Applied Mathematics, National University of Kaohsiung)
- Co-organizer: Shih-Wei Chou (Department of Financial Engineering and Actuarial Mathematics, Soochow University)
Program & Speakers
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On the existence and the multiplicity of vector solutions to nonlinear Schrödinger equations with three wave interaction
Tomoharu Kinoshita (Waseda University)
In this talk, I would like to talk about the nonlinear Schrödinger equations with three wave interaction. For large interaction, Pomponio (2010) showed that there exists a ground state which is a vector solution. On the other hand, for small interaction the existence of vector solutions is not even known. The purpose of this talk is to establish the existence and the multiplicity of vector solutions of the system for small and large three wave interaction.
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Semiclassical States in Hartree–Fock Type Systems with Interspecies Repulsion
Ying-Chieh Lin (National University of Kaohsiung)
In this talk, we consider a class of Hartree–Fock type systems consisting of two coupled nonlinear Schrödinger equations with Coulomb interaction terms. We focus on the regime where the interaction between different components is repulsive. We establish the existence and multiplicity of positive semiclassical solutions and analyze how these solutions are influenced by the structure of the underlying potential functions, in particular their minima. Special attention is devoted to the concentration behavior of solutions in the semiclassical limit, where we identify the locations and profiles of concentrating states. In addition, we discuss the variational characterization of these solutions and examine whether they correspond to least energy states. The results reveal a rich interplay between repulsive interactions and concentration phenomena, leading to qualitative differences from the attractive case.
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The System of Nonlinear Schrodinger Equations with Linear Coupling
冠祥 王 (National Kaohsiung Normal University)
In this talk, we will undertake a comprehensive study for a system of nonlinear Schrodinger equations with linear coupling. The type of systems arises in the study of nonlinear optics. We will address questions related to local and global well-posedness, fintie time blowup, and scattering in the energy space.
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Global Transonic Entropy Solutions of Fanno-Rayleigh Flow Through Time Dependent Nozzles
ShihWei Chou (Soochow University)
In this talk we investigate the initial boundary value problem for the one-dimensional compressible Euler equations with friction and heat transfer effect, a model describing Fanno-Rayleigh flow in a time dependent, variable area nozzle. We identify a sharp threshold expressed in terms of the friction coefficient and the geometric properties of the duct that admit the existence of transonic solutions. In addition, we derive explicit lower bounds for both the Mach number and the power flux of the gas, through a nonlinear dilation of the characteristic fields; these bounds allow us to determine whether a feasible solution remains purely supersonic or make a transition to transonic flow. To conduct our investigation, an innovative variant of the generalized Glimm scheme is developed.
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Global Existence for Stochastic Shallow Water Equations with irregular topography and Hurst-dependent Noise
Po-Chih Huang (National Chung Cheng University)
In this talk, we consider initial-boundary value problem for shallow water model with stochastic source term. We propose a generalized Glimm scheme. We construct approximate solutions for both interior and boundary Riemann problems, carefully estimating the corresponding residuals to ensure consistency. By introducing a modified Glimm functional that accounts for wave interactions and stochastic perturbations, we prove that the total variation of the approximate solutions remains uniformly bounded, yielding the global existence of entropy solutions. Furthermore, by exploring the evolutionary dynamics of the stochastic waves, we derive physically meaningful dissipative initial and boundary conditions that effectively control non-physical oscillations.
MS31 Low-Rank Tensor Approximation and Neural Sampling Methods for High-Dimensional Scientific Computing Main organizer: Kejun Tang (Great Bay University) · 7 talks
Organizers
Main organizer: Kejun Tang (Great Bay University)
- Co-organizer: Qifeng Liao (ShanghaiTech University)
Program & Speakers
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Weak TransNet: a Petrov-Galerkin based method for partial differential equations
Zhihang Xu (Shanghai Institute of Technology)
We propose Weak TransNet (WTN), a Petrov–Galerkin method for solving PDEs with singular solutions or without strong formulations. WTN combines weak formulations with TransNet, a shallow neural network that builds a predetermined neural feature space. The PDE solution is approximated as a linear combination of these features, with coefficients obtained by minimizing a weak-form loss. By decoupling basis construction from coefficient computation, WTN alleviates non-convexity and ill-conditioning in neural training. Numerical experiments demonstrate its robustness and efficiency.
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Physics-Informed Laplace Neural Operator: Data-Efficient and Out-of-Distribution Robust PDE Surrogate Modeling
Heechang Kim (POSTECH)
Neural operators provide fast surrogate solvers for parametric PDEs, but purely data-driven training often requires large datasets and can be unreliable in small-data or out-of-distribution settings. In this talk, I will present the Physics-Informed Laplace Neural Operator (PILNO), which extends the Laplace Neural Operator by enforcing PDE, boundary condition, and initial condition losses during training. PILNO also uses unlabeled “virtual” inputs for physics-only supervision and temporal-causality weighting to emphasize early-time dynamics. Benchmarks on Burgers’ equation, Darcy flow, reaction-diffusion, and forced KdV show that PILNO improves accuracy in small-data regimes and generalizes better beyond the training distribution.
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Solving high-dimensional partial differential equations with deep learning requires rethinking sample generation
Kejun Tang (Great Bay University)
This work studies the role of random collocation points in deep learning methods for solving high-dimensional partial differential equations. Although deep neural networks have shown remarkable performance in data-driven methods, the selection of random collocation points remains a critical issue in solving high-dimensional partial differential equations (PDEs). From the perspective of error analysis, random sampling introduces statistical errors into the discretization of the loss functional, and the geometric properties of high-dimensional spaces may cause these statistical errors to dominate the final approximation error. We propose a new min-max formulation that simultaneously optimizes the approximate solution and the training samples. The main idea is to adjust the distribution of random samples such that the residual induced by the neural network maintains a nearly uniform profile during training.
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Generative Modelling and Probability Flow : from A to B
Xiang Zhou (City University of Hong Kong)
In this talk, I will demonstrate the impact of the generative modelling to some longstanding challenges in computational mathematics and statitical physics and new opportunities for scientific computing. The specific topics are (1)How weak form of the PDE signicicantly reduces the l cost of flow-based transport map (normalizing flow) to solve Kolmogorov forward equation (2) How the perpsective of probability flow equivalence defines the Levy score function for the non-Gaussian Levy-Ito SDE, and its application to Onsager-Machlup functional and transport-based particle algorithm and in stochastic thermodynamics for non-Gaussian active matter. (3) How the flow-based generative method solves the gradient flow of free energy in Wasserstein space in geometric Dawson-Gartner framework. : 2604.11519; 2504.06628 [PRL 2026]; 2412.19520 Levy Score Function [SINUM2026]; 2409.01340[SIAP2025]; 2405.19256 : Weak Generative Sampler[SISC2026]; 2306.02063 [NeurIPS 2023].
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Low-rank analysis of the inverse of the discretized Laplace operator
Chuanfu Xiao (Xiangtan University)
Solving high-dimensional partial differential equations often leads to extremely large structured linear systems, for which direct matrix inversion is computationally prohibitive. Tensor representations provide an efficient framework for compressing and manipulating such large-scale operators. A fundamental question is whether the inverse of a structured matrix, arising from the discretization of a differential operator, can still be accurately represented in a low-rank tensor format. In this talk, we investigate the low-rank structure of the inverse of discretized Laplace operator. We prove that it admits an accurate low-rank TT/QTT approximation. This result provides both a theoretical explanation for the low-rank behavior of inverse operators and practical guidance for designing efficient numerical algorithms based on tensor representations.
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Multiscale model reduction based on local low-rank approximations
Chupeng Ma (Great Bay University)
I will present a multiscale model reduction method based on local low-rank approximations for elliptic PDEs with heterogeneous coefficients. The low-rank approximations are built from the singular value decomposition of a restriction operator defined on generalized harmonic spaces. Rigorous convergence analysis of the method as well as numerical experiments will be provided.
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Model Compression via Dynamic System
Fenglei Fan (Cornell)
Recently, the escalating demand for memory and computational resources by large models has presented formidable challenges for their deployment in resource-constrained environments. One prevalent way to serve large models is to develop effective model compression approaches that crop the size of large models while maintaining acceptable performance levels. In this talk, we introduce a novel and general-purpose approach, referred to as hyper-compression, that redefines model compression as a problem of parameter representation. Specifically, we extend the concept of hypernets into what we term a ‘hyperfunction'. Then, the hyperfunction is designed based on dynamic systems.
MS32 Uncertainty Quantification and Mathematical Foundations of Deep Learning: Recent Advances in Methods, Theory, and Applications Main organizer: Jiahua Jiang (ShanghaiTech University) · 7 talks
Organizers
Main organizer: Jiahua Jiang (ShanghaiTech University)
- Co-organizer: Ling Guo (Shanghai Normal University)
- Co-organizer: Liang Yan (Southeast University)
Program & Speakers
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A warm-basis method for bridging learning and iteration: a case study in fluorescence molecular tomography
Jiahua Jiang (ShanghaiTech University)
Fluorescence Molecular Tomography (FMT) is non‑invasive but depth reconstruction remains challenging. Conventional iterative methods have poor z‑resolution even with advanced regularization, while supervised learning improves accuracy but requires impractical large paired datasets. This raises the question of how to combine learning with iterative schemes. We propose a warm‑basis iterative projection method (WB‑IPM) with theoretical foundations. It significantly outperforms both learning‑based and iterative approaches, and uses a weaker loss function based only on directional differences, reducing training effort. These benefits are supported by error analysis and experiments on simulated and real data.
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Functional Multi-Target Detection via Bispectrum Inversion
Ruiyi Yang (Shanghai Jiao Tong University)
We develop a functional theory for multi-target detection, recovering a compactly supported signal from a single noisy observation containing many unknown translations. Our model permits continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond standard discrete, grid-aligned white-noise settings. We study two uninitialized autocorrelation-based algorithms. Both first estimate the signal bispectrum using a debiased third-order empirical autocorrelation, then recover the signal either by a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both methods, we establish non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The error bounds depend on signal smoothness and bispectrum estimation accuracy, which is governed by the noise structure and number of signal occurrences. Numerical experiments support the theory and show accurate recovery in low-SNR regimes.
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An Efficient Reduced-Order Model Based on Dynamic Mode Decomposition for Parameterized Spatial High-Dimensional PDEs
Zhen Gao (Ocean University of China)
Accurately constructing a reduced order model (ROM) of parameterized partial differential equations (PDEs) has always been a challenging problem in engineering and applied sciences. Dynamic mode decomposition (DMD) is a popular data-driven method and proposed for the ROM of time-dependent problems. However, it is invalid for the parameterized problems. In this talk, we will first extend the DMD to the parameterized problems and then introduce recent development. We apply the proposed methods to various parameterized PDEs. The results demonstrate the applicability and efficiency of the proposed ROMs. Furthermore, the proposed ROMs show better predictive ability than the POD-based ROMs outside of the training time region.
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Flow-based generative models for amortized Bayesian inference in regression and inverse PDE problems
Sahoqian Zhou (Huazhong University of Science and Technology)
Bayesian inference enables uncertainty quantification in scientific ML, but conventional methods solve a new inference problem per observation set, limiting real-time use in monitoring and digital twins. Many problems involve function-space inference with varying observation counts/locations, challenging amortized methods. We propose Flow-ABI, a flow-based framework for amortized Bayesian inference in regression and inverse PDE problems: (i) a functional prior model learning function-space priors via flow matching, and (ii) a set-conditioned posterior sampler mapping observations to functional posteriors. The model handles varying observation sets, is permutation-invariant, and generalizes across discretizations. Once trained, it enables near-real-time posterior sampling without retraining, and integrates with physics-informed neural networks and neural operators. Experiments show Flow-ABI captures Gaussian/non-Gaussian posteriors with speedups over two orders of magnitude versus HMC.
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Sharp Sobolev Sandwich and Approximation Rates of Shallow ReLU$^k$ Networks
Juncai He (Tsinghua University)
We present a Radon-domain framework for analyzing shallow ReLU$^k$ networks, connecting Barron-type spaces, Hilbert integral representations, and $L^p$ Sobolev regularity. The main result is a sharp Sobolev sandwich for $L^p$ ridge integral spaces, with sharpness explained by microlocal analysis. We also derive improved high-probability approximation rates for linearized/random feature networks and illustrate them numerically.
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Deep learning-enhanced reduced-order ensemble Kalman filter for efficient Bayesian data assimilation
Yanyan Wang (Shanghai Normal University)
Bayesian data assimilation for parametric PDE systems is computationally expensive because it requires repeated forward model evaluations. We present a deep learning-enhanced reduced-order ensemble Kalman filter for Bayesian data assimilation in parametric PDE systems. The proposed method combines operator inference with a data-driven model-error correction strategy. A reduced-order model is first learned from long-term coarse-grid simulations to capture the dominant system dynamics. A neural network is then trained to correct the discrepancy between the reduced-order and high-fidelity models. The learned correction is incorporated into the online ensemble Kalman filtering procedure, improving estimation accuracy while maintaining low computational cost. Numerical results demonstrate that the proposed framework provides an effective balance between efficiency and accuracy for nonlinear data assimilation problems.
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Multigrid Neural Operators for Helmholtz Preconditioning and Diff-ANO Guided UCST Inversion
Xinliang Liu (KAUST)
We introduce the Multigrid Neural Operator (MgNO) and the Multigrid Neural Preconditioner (MgNP), a unified framework that integrates classical multigrid methodology with modern deep learning to address complex problems. First, we present a concise neural operator architecture, drawing an analogy to fully connected neural networks. To overcome spectral bias in operator learning, we extend this design into MgNO, built on the MgNet architecture, where multigrid V-cycle structure is embedded. Next, we develop a multigrid-inspired neural preconditioner for challenging problems such as high-wavenumber Helmholtz equations. This preconditioner learns both smoothing and coarsening operators from data. The resulting multichannel neural preconditioner achieves substantial convergence speedups compared to classical alternatives. Finally, we demonstrate the robustness and efficiency of the proposed approaches in applications to ultrasonic computed tomography.
MS33 Geometric Shape Generation (II) Main organizer: Joseph Cho (Handong Global University) · 4 talks
Organizers
Main organizer: Joseph Cho (Handong Global University)
- Co-organizer: Seong-Deog Yang (Korea University)
- Co-organizer: Masashi Yasumoto (Tokushima University)
Program & Speakers
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Backlund transformations for curves and their applications to semi-discrete K-surfaces
Shogo Shimada (Tokushima University)
Transformation theory of surfaces plays an important role in the study of integrable systems. We first focus on Backlund transformations for surfaces with constant negative Gaussian curvature (K-surfaces). Backlund transformations provide new K-surfaces from a given one and, at the same time, generate new solutions to the sine-Gordon equation - the structure equation of K-surfaces - from known ones. We then introduce Backlund transformations for curves with constant nonzero torsion. We show that these transformations give rise to two types of semi-discrete integrable equations, generalizing results obtained by Calini and Ivey. Furthermore, we derive superposition principles for solutions of these equations through the geometry of semi-discrete K-surfaces. Finally, we present examples of semi-discrete K-surfaces constructed via these superposition principles. This is joint work with Masashi Yasumoto.
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Iterated minimal Darboux transformations
Takeshita Sakuma (Tokushima University)
Transformations for surfaces are crucially important in constructing new surfaces. Here we focus on isothermic surfaces. They admit conformal curvature line coordinates, and it is known that they admit various transformations. In this presentation, we introduce the permutability of transformations for isothermic surfaces in the Euclidean space. We show that a superposition principle of Darboux transformations for minimal surfaces can be viewed as an application of the permutability of transformations for isothermic surfaces. Furthermore, we illustrate concrete examples of minimal surfaces obtained by our result, and explore their asymptotic behaviors. This is joint work with Masashi Yasumoto.
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Discrete maximal surfaces from incircular nets
Denis Polly (TU Wien)
Incircular nets have been used in the statistical mechanics community to study the Ising model. Surprisingly, it turned out that a certain lift of an incircular net in Lorentz space converges to a maximal surface as the model approaches its thermodynamic lift. This motivates the question, which of these lifts, interpreted as discrete surfaces, can already be considered maximal in the discrete category. In this talk, we will describe a subclass of isothermic incircular nets that lift to (discrete) S-isothermic surfaces. Translating an approach introduced by Bobenko, Hoffmann and Springborn (for minimal surfaces in Euclidean space), we will then identify a subclass of isothermic incircular nets that correspond to discrete maximal surfaces and their associated families. This talk is based on joint work with Niklas Affolter, Felix Dellinger, Christian Müller and Nina Smeenk
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Bistable kinematic structures from integrable discrete surfaces
Gudrun Szewieczek (University of Innsbruck)
Bistable, or “snapping,” structures have the ability to hold two stable states without continuous energy input. This talk introduces a novel class of such systems: snapping four-bar nets. These are multi-looped kinematic chains that possess exactly two distinct assembly configurations and zero degrees of freedom. A transition between these states is not a rigid-body motion but a “snap-through” enabled by material flexibility. Central to this talk is an unexpected bridge between kinematics and discrete differential geometry. We demonstrate how these bistable structures can be systematically designed using discrete Koenigs nets—a broad class of integrable discrete surfaces that includes, among others, discrete minimal surfaces. This is joint work with Daniel Huczala, Martin Pfurner and Hans-Peter Schröcker.
MS34 High-Order Numerical Methods for Hyperbolic Conservation Laws Main organizer: Jiaxi Gu (Pohang University of Science and Technology) · 4 talks
Organizers
Main organizer: Jiaxi Gu (Pohang University of Science and Technology)
- Co-organizer: Bao-Shan Wang (Ocean University of China)
Program & Speakers
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Foliation Structures and Numerical Validation of Scalar Hyperbolic Conservation Laws on Manifolds
Bao-Shan Wang (Ocean University of China)
This talk presents a theoretical and computational framework for scalar (nonlinear) hyperbolic conservation laws on closed regular 2D manifolds (e.g., sphere and torus), demonstrating how kernel-style embedding techniques, specifically the Closest Point Method, can be effectively combined with high-order numerical schemes to solve surface PDEs with shocks on complex geometries. A geometry-compatible flux formulation is introduced, which induces a natural foliation of the manifold forming regular leaves, singular points, and separatrices. The foliation decomposes the surface PDEs into a family of 1D conservation laws along the leaves, enabling global flow dynamics to emerge from their collective evolution, as validated by 2D-1D reduction tests. The Closest Point Method coupled with high-order WENO scheme (cp-WENO) is employed to simulate the hyperbolic equations on both spherical and toroidal geometries, confirming the geometry-induced structures.
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A data-driven third-order WENO schemes based on conservative approximation to the derivative
Kwanghyuk Park (POSTECH)
We propose a feedforward neural network based on a conservative approximation to the derivative from point values for WENO schemes applied to hyperbolic conservation laws. The network replaces the classical WENO weighting procedure by mapping three-point stencil values to two nonlinear weights. For supervised training, we construct a labeled dataset in which numerical fluxes are determined so that their differences provide high-order approximations to the derivative. The loss function encourages the neural network to satisfy ENO property and the symmetry shared by WENO3-JS and WENO3-Z. The resulting WENO3-CADNN schemes generalize robustly across benchmark problems and grid resolutions, outperforming WENO3-Z.
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Structure preserving unfitted discontinuous Galerkin method for hyperbolic conservation laws
Pei Fu (Nanjing University of Aeronautics and Astronautics)
This talk will present a family of high order cut finite element methods based on the discontinuous Galerkin (DG) framework for hyperbolic problems. To avoid the small-cut-cell issue, a ghost penalty stabilization is employed to stabilize the scheme. The strong stability-preserving Runge–Kutta method is used for time discretization, in which the time step is chosen independently of the size of the cut elements. We show that the proposed methods possess stability and accuracy properties as the standard DG methods on fitted meshes. We also prove and verify that the scheme preserves the bound-preserving property for scalar conservation laws when a flux limiting work is applied. Finally, numerical experiments will be given to demonstrate that the proposed methods achieve high-order accuracy for smooth solutions and perform well for problems with discontinuities.
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A modified WENO-Z+ scheme based on the unnormalized nonlinear weights of the rewritten fifth-order adaptive order WENO scheme
Li Yuan (AMSS)
This paper proposes a modified fifth-order WENO-Z+ type scheme by comparing the unnormalized weights of the rewritten fifth-order adaptive-order (AO) WENO scheme with those of the existing WENO-Z+ scheme. The fifth-order WENO-AO scheme using sub-stencils of unequal sizes is reformulated into the standard WENO combination form with sub-stencils of equal size, and it is found that its unormalized weights can be decomposed into three components: a constant term, a local smoothness measurement term for each sub-stencil, and a global smoothness measurement term for the global stencil. To fully utilize the latter two terms to construct an improved WENO scheme, an adaptive scaling factor is introduced to balance the contributions of the second term and the modified third term. Numerical experiments demonstrate that the improved fifth-order WENO-Z+ scheme can achieve high resolution in smooth regions while capturing shock discontinuities sharply and robustly.
MS35 Fast Solvers for Multiphysics Problems Main organizer: Shihua Gong (The Chinese University of Hong Kong, Shenzhen) · 8 talks
Organizers
Main organizer: Shihua Gong (The Chinese University of Hong Kong, Shenzhen)
- Co-organizer: Wei Wang (Academy of Mathematics and Systems Science, CAS)
- Co-organizer: Chupeng Ma (Great Bay University)
Program & Speakers
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A Second-order Structure-preserving Parametric FEM for surface evolution
Beiping Duan (Shenzhen MSU-BIT University)
We present a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and the volume-preserving mean curvature flow. We first reformulate the original evolution equations into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the unknown evolving surface and drives points on the evolving surface to move in their tangent spaces so as to reduce the associated harmonic energy. As a result, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface, unless singularities occur. The volume-preserving property is theoretically guaranteed by the delicate design of the scheme, while energy dissipation is enforced through a Lagrange multiplier. The proposed framework can be readily extended to other geometric flows.
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A Pressure-Robust Nonconforming Immersed Finite Element Method for Stokes Interface Problems
FENG WANG (Nanjing Normal University)
It is well-established that an appropriate modification of test functions may lead to pressure-robust mixed methods for the Stokes problems. However, for immersed finite element approximations of the Stokes interface problems on unfitted meshes, it is remains unclear whether the error of the velocity is independent of the pressure, since the velocity and the pressure are cou- pled in one of the interface conditions. In this talk, we provide a positive answer by a novel decomposition of the discontinuous pressure to a continuous function and a velocity related discontinuous part. We demonstrate that the immersed Crouzeix-Raviart/P0 element method achieves pressure robustness via an H(div)-conforming reconstruction of the test functions on the right-hand side. The stability and optimal error estimates of the proposed method are established with constants independent of the interface position relative to the mesh. Numerical experiments are presented to validate the theoretical findings.
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Strict Constraint Satisfaction in Multiphysics Solvers via Convexity-Driven Training
Shuo Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
In the development of machine learning-based solvers for multiphysics problems, constraints are typically incorporated into the optimization objective as penalty terms. However, this approach often yields solutions that cannot strictly satisfy the underlying physical constraints, leading to non-physical results. In this work, we propose a training method that ensures strict constraint adherence and provide a theoretical analysis of its applicability. Distinct from general constrained optimization, the proposed objective function exhibits a specific form of convexity, which mitigates training difficulties and accelerates convergence. Numerical experiments on several computational instances demonstrate the effectiveness of this strategy. This is a joint work with Wenyu Dong.
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Unified convergence analysis of non-variational finite element methods for a Pucci equation in two dimensions
Zhiyu Tan (Xiamen University)
In this paper, a unified framework based on discrete Miranda–Talenti type inequalities is developed for the design and analysis of non‑variational finite element methods for a Pucci equation in two dimensions. The proposed framework is applicable to a wide range of discretizations, including conforming finite element methods, $C^0$ interior penalty methods, non‑conforming finite element methods, and discontinuous finite element methods. A unified error analysis is carried out, from which quasi‑optimal convergence results are obtained. To solve the resulting non‑smooth discrete problem, a fixed‑point method is proposed. The convergence of this method is rigorously established, and an explicit convergence rate is provided. Numerical experiments are presented and confirm the theoretical findings.
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Sharp Inf-Sup Estimate and Fast Solvers for the Stokes Equation in Tight Domains with Periodic Pillars
Shihua Gong (The Chinese University of Hong Kong, Shenzhen)
Microfluidic chips are widely used to separate blood cells and other particles by arranging periodic pillar arrays inside flow channels. However, as the pillar density increases, standard numerical solvers for the Stokes equations often suffer from severe convergence stagnation. In this work, we show that this difficulty is not merely algorithmic, but is rooted in the geometry-induced degradation of the pressure–velocity coupling stability. For tight domains with periodic pillars, we prove that the continuous Ladyzhenskaya–Babuška–Brezzi (inf-sup) constant decays sharply as $\mathcal{O}(m^{-1})$, where $m$ denotes the number of pillars per unit length. This estimate implies an $\mathcal{O}(m)$ amplification in finite element a priori error bounds and an $\mathcal{O}(m^2)$ growth in the condition number of the pressure Schur complement. In this talk, I will also discuss several preconditioning techniques for efficiently solving the resulting Schur complement system.
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Robust Numerical Methods for Some Topology Optimization Problems
Huangxin Chen (Xiamen University)
In this talk, we will introduce a prediction-correction iterative convolution thresholding method for the topology optimization for heat transfer problem. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. The application of the new method for the coupled thermal-fluid topology optimization will be discussed. We will also introduce a robust phase filed method for structural topology optimization.
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Adaptive Local Multilevel Methods for Elliptic and Maxwell Eigenvalue Problems
Qigang Liang (Tongji University)
In this talk, we propose an efficient adaptive local multilevel preconditioned Jacobi-Davidson method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson correction equation arising from adaptive finite element methods. As a result, the algorithm holds optimal computational complexity $O(N)$. The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, we also extend this method with Helmholtz projection to solve Maxwell eigenvalue problems. Numerical experiments on complex domains are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method. This is a joint work with Prof. Xuejun Xu and Ph.D. students (Jianing Guo and Qingquan Zhang).
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An isogeometric fast multipole boundary integral method for solving multiple scattering problems
BO WANG (Hunan Normal University)
This talk presents a multi-patch isogeometric fast multipole boundary element method for simulating multiple scattering problems. The method preserves exact geometry of complex scatterers using NURBS patches, which also serve as the basis for discretizing the boundary integral equation. Exploiting the local support of NURBS, we recast the dense matrix-vector product as a quadrature-source potential evaluation and accelerate it via the fast multipole method. To enhance iterative solver convergence, we design a scatterer-aware ACA-SMW preconditioner that retains complete coarse-level scatterer blocks and performs DOF bisection only within individual scatterers. Numerical results confirm high-order accuracy for smooth geometries, improved GMRES convergence, and near-linear scaling for large-scale problems with complex non-convex scatterers.
MS36 Machine Learning Methods for Stochastic Dynamics Main organizer: Li Zeng (Fuzhou University) · 4 talks
Organizers
Main organizer: Li Zeng (Fuzhou University)
- Co-organizer: Xiaodong Feng (Beijing Normal-Hong Kong Baptist University)
Program & Speakers
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Identifiable learning of dissipative dynamics
Aiqing Zhu (National University of Singapore)
Complex dissipative systems appear across science and engineering, from polymers and active matter to learning algorithms. These systems operate far from equilibrium, where energy dissipation and time irreversibility govern their behavior but are difficult to quantify from data. In this work, we introduce a universal and identifiable neural framework that learns dissipative stochastic dynamics while ensuring interpretability, expressiveness, and uniqueness. Our method identifies a unique energy landscape, separates reversible from irreversible motion, and allows direct computation of the entropy production, providing a principled measure of irreversibility and deviations from equilibrium. Applications to polymer stretching in elongational flow and to SGLD reveal new insights, including super-linear scaling of barrier heights and sub-linear scaling of entropy production rates with the strain rate, and the suppression of irreversibility with increasing batch size.
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Flow-based joint Bayesian state and parameter inference for nonlinear stochastic dynamical systems
Xintong Wang (Shanghai Normal University)
Bayesian state and parameter inference for nonlinear stochastic dynamical systems is fundamental but challenging across science and engineering. Gaussian filters struggle with non-Gaussian distributions, while sequential Monte Carlo methods are costly and suffer from particle degeneracy in high dimensions. Although deep generative approaches can model complex distributions, they still face difficulties in balancing accuracy and computational efficiency. To address these challenges, we develop a unified framework built upon the flow-based Bayesian filter, which integrates normalizing flows to construct a novel latent linear Gaussian state-space model. This framework enables efficient data-driven filtering, smoothing, and parameter sampling without requiring explicit knowledge of system dynamics and observation models. Numerical experiments show that the proposed method achieves higher accuracy and efficiency than existing approaches.
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Flow-Based Adaptive Density Approximation for Fokker--Planck Type Equations
Li Zeng (Fuzhou University)
Fokker--Planck type equations describe the evolution of probability densities arising from stochastic dynamical systems, but their numerical solution remains challenging in high dimensions, on unbounded domains, and in the presence of nonlocal operators. In this talk, we present flow-based methods for adaptive density approximation of such equations. The main idea is to represent the unknown density by an invertible neural transformation, so that positivity and mass conservation are naturally encoded. We first discuss temporal normalizing flows for time-dependent Fokker--Planck equations, and then describe extensions to fractional Fokker--Planck equations with nonlocal operators on unbounded domains. Finally, we introduce B-KRnet, a bounded-domain normalizing flow, and discuss its applications to density estimation and PDE-based density approximation. Together, these works demonstrate the effectiveness of flow-based neural models for probability-density evolution equations.
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Operator Learning for Solving Fokker-Planck Equations with Various Initial Conditions
Yaobin Wang (Beijing Normal-Hong Kong Baptist University)
The Fokker-Planck equation (FPE) describes the evolution of probability density functions for stochastic dynamical systems. We propose a conditional normalizing-flow physics-informed neural network (PINN) for approximating the FPE solution operator over a family of initial conditions. Using the Chapman-Kolmogorov equation, we reformulate the task as learning the transition density from a Dirac mass at an arbitrary initial point. To avoid the singular map induced by the Dirac initial distribution, we use the density of a linearized stochastic differential equation (SDE) as the flow base distribution, which gives an accurate small-time approximation to the target density. We further introduce a time-weighted residual loss to reduce small-time instability while preserving the causal structure of the dynamics. Numerical experiments demonstrate the accuracy, robustness, and efficiency of the proposed method.
MS37 Numerical Methods for Phase-Field Models: Advances, Challenges and Applications Main organizer: Chaoyu Quan (The Chinese University of Hong Kong (Shenzhen)) · 4 talks
Organizers
Main organizer: Chaoyu Quan (The Chinese University of Hong Kong (Shenzhen))
- Co-organizer: Jiang Yang (Southern University of Science and Technology)
- Co-organizer: Wei Zhang (Beijing Normal-Hong Kong Baptist University)
Program & Speakers
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Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using General Multipliers
Chaoyu Quan (The Chinese University of Hong Kong)
A unified framework is proposed to establish the energy dissipation of implicit-explicit linear multistep methods (IMEX-LMMs) applied to gradient flows. Given a fixed IMEX-LMM, it is shown that finding an appropriate multiplier for constructing the dissipative energy can be formulated as a feasibility problem over the multiplier coefficients, where the involved positivity inequalities on the unit circle are relaxed to a finite-dimensional linear programming problem after discretization. Within this framework, two specific multipliers are obtained to establish the energy dissipation of the sixth-order IMEX backward differentiation formula (IMEX-BDF6) and a seventh-order IMEX weighted and shifted BDF, and a new eighth-order energy-dissipative IMEX-LMM is developed together with its multiplier. To the best of our knowledge, these are the first energy-dissipation results for the IMEX-BDF6 method and the IMEX-LMMs of order higher than six.
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Efficient Structure-Preserving Exponential Time Differencing Schemes for the Gradient Flow of Q-tensor Model
Zhonghua Qiao (The Hong Kong Polytechnic University)
The Landau–de Gennes Q-tensor theory provides a fundamental framework for modeling nematic liquid crystals. Numerical simulations of the corresponding Q-tensor gradient flows are challenging due to the presence of stiff nonlinear terms and the need to preserve essential physical constraints, in particular the energy dissipation law and the maximum bound principle (MBP). In this work, we propose a class of first- and second order stabilized exponential time differencing (ETD) schemes for the Q-tensor flow. We rigorously prove that the proposed schemes unconditionally preserve the discrete MBP and the original energy dissipation law. Moreover, optimal-order error estimates are established for both schemes. Numerical experiments are presented to validate the theoretical results, demonstrating that the proposed schemes achieve high-order temporal accuracy while strictly preserving the physical bounds and energy dissipation property, even for relatively large time step sizes.
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Length- and Energy-Preserving Numerical Schemes for Projected Gradient Flows of Vector Fields
Jiang Yang (Southern University of Science and Technology)
Developing numerical schemes that simultaneously preserve length constraints energy dissipation for projected gradient flows of vector fields is a long-standing challenge, particularly for high-order or decoupled formulations. In this talk, we present a novel structure-preserving numerical framework based on a newly developed class of product-type implicit-explicit Runge-Kutta (PRK) methods. For unit vector fields with Dirichlet energy, we propose a linear, second-order time-stepping scheme that unconditionally guarantees both unit-length preservation and original energy dissipation without requiring computationally expensive nonlinear algebraic solvers. To the best of our knowledge, this represents the first second-order linear scheme to achieve dual structure preservation for harmonic map heat flows. Furthermore, we discuss the extension of this framework to more general nonlinear potentials and complex manifold constraints using the Scalar Auxiliary Variable (SAV) approach.
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Efficient spectral methods for solving incompressible Navier-Stokes equations in complex domains
Jie Shen (Eastern Institute of Technology, Ningbo)
In this talk, we will present an efficient, fully decoupled, divergence-free spectral method for the incompressible Navier–Stokes equations in two-dimensional complex domains. The method combines a circular embedding technique with a pressure reconstruction strategy. The physical domain is embedded into a disk, where the geometric boundary constraints are separated from the governing equations through an appropriate decomposition. The pressure is then decomposed into a particular component determined by the forcing term and a homogeneous component whose coefficients act as Lagrange multipliers enforcing the incompressibility constraint. Numerical experiments demonstrate the spectral accuracy, computational efficiency, and robustness of the proposed method in complex geometries.
MS38 Multiscale modeling and numerical methods for multiphysics materials Main organizer: Luchan Zhang (Shenzhen University) · 8 talks
Organizers
Main organizer: Luchan Zhang (Shenzhen University)
- Co-organizer: Xiang Yang (Hong Kong University of Science and Technology)
- Co-organizer: Qin Xiaoxue (Shanghai Univeristy)
Program & Speakers
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Finite element method for the spin-diffusion Landau-Lifshitz-Gilbert (SDLLG) model
Weiying Zheng (Chinese Academy of Sciences)
In this talk, I will present the finite element approximation of the nonlinear spin-diffusion Landau-Lifshitz-Gilbert (SDLLG) model. We first establish the unconditional energy stablity for the continuous LLG model independent of the small damping parameter. A fully-discrete finite element method is proposed to solve the LLG model. Unconditional stability for the numerical scheme is proved with respect to the damping parameter. Based on the analysis for the LLG model, we further prove the unconditional stability for the SDLLG model under a very weak and physically reasonable assumption. A decoupled time-marching scheme is proposed for the coupled SDLLG problem.
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Mixed scheme about ROM/FOM with space-adaptive partitioning strategy for quasi-static peridynamic fracture simulation
Yufeng Nie (Northwestern Polytechnical University)
In this talk, we present a mixed reduced-order/full-order model (ROM/FOM) scheme based on space-adaptive partitioning strategy for peridynamics (PD) to efficiently simulate brittle fractures. Generally, the drastical displacement mutation occurs around the crack tip and path, resulting in the ROM more sensitive to high-fidelity snapshots, and even leading to failure of ROM. To overcome this difficulty, the entire computational domain is partitioned into two types of regions: the damage regions are simulated using full-order PD solver, while the sub-system over remaining regions is projected onto low-dimensional subspaces obtained by the proper orthogonal decomposition (POD) method. The domain partitioning is adaptively performed based on bond-breakage data and no a priori knowledge of cracks is required.
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Reconstruction of phase field model with an EnKF-based restoration framework
Yibao Li (Xi'an Jiaotong University)
Incomplete or damaged regions disrupt the consistency of numerical simulations, making both analysis and prediction challenging. In this study, we present an efficient and flexible restoration framework of reconstructed solution for phase field model. To address this challenge, the Ensemble Kalman Filter (EnKF) data assimilation method is employed to estimate optimal model parameters with partial observations sampled from the target phase field states to be reconstructed. The estimated parameters are then incorporated into the governing model to reconstruct the missing morphology. A series of twin experiments are conducted to systematically evaluate the effectiveness and robustness of the method under controlled settings. The results confirm that the EnKF-based restoration framework is capable of reliably restoring structures even with complex reconstructed domains.
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3D Continuum Modeling and Simulation of Low-Angle Grain Boundaries
Xiaoxue Qin (Shanghai University)
We have developed a range of continuum models that describe the structure and dynamics of low angle grain boundaries in three dimensions. The static model minimizes grain boundary energy according to Frank's formula constraints, while the dynamic model includes the motion and reactions of dislocations. We have developed numerical algorithms based on the projection method. In comparing our simulation results with atomistic simulations, our model accurately predicts the dislocation structure, energy profiles of grain boundaries, and their dynamics. These findings highlight the potential of continuum models to enhance the understanding of material behavior and solid mechanics.
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A unified variational model for grain boundary dynamics incorporating microscopic structure
Yang Xiang (HKUST)
Recent experiments, atomistic simulations, and theoretical predictions have identified various new types of grain boundary motions that are controlled by the dynamics of underlying microstructure of line defects (dislocations or disconnections), to which the classical motion by mean curvature model does not apply. We propose a unified variational framework to account for all the underlying line defect mechanisms for the dynamics of both low and high angle grain boundaries and the associated grain rotations. The variational formulation is based on the developed constraints of the dynamic Frank-Bilby equation that governs the microscopic line defect structures. The unified variational framework is more efficient to describe the collective behaviors of grain boundary networks at larger length scales. It also provides a mathematically tractable basis for rigorous analysis of these partial differential equation models and for the development of efficient numerical methods.
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Stacking pattern of short-range order units in binary metallic glasses: A hypergraph approach
Jingli Ren (Zhengzhou University)
This study investigates stacking patterns between short-range order (SRO) units in three binary metallic glasses by establishing a framework combining hypergraph and molecular dynamics simulation. It is found by maximal clique decomposition that SRO unit at 300 K stacking primarily consists of ternary and quaternary maximal cliques, collectively accounting for about 90% of the total. An exponential relationship between maximal clique population and average atomic radius (ln Nmc=αγ+C) is established, where the absolute value of α is related to atomic radius ratio. We report a close correlation between the dynamic behavior of Nmc and the glass transition in metallic glasses. A logistic growth model is presented to reveal the evolution of Nmc from initial unlimited growth to hindered growth, with this transition initiating at about 1.16 Tg.
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A new phase-field model for martensitic phase transitions driven by material forces
Peicheng Zhu (Shanghai University)
We present the existence of weak solutions to an IBVP for a new model of structural phase transitions, in which the order parameter is conserved. Two types of weak solutions will be stated, i.e., usual weak solution (defined by employing the technique of integration by parts), and weak solution defined by combining the usual concept of weak solution and the notion of viscosity solutions. Applications include: To describe martensitic phase transitions in, e.g., Shape Memory Alloys. Numerical results will also be presented.
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Stochastic Peierls–Nabarro Modeling for Predicting Strength-Optimized Compositions of High-Entropy Alloys
Luchan Zhang (Shenzhen University)
High-entropy alloys (HEAs) exhibit outstanding mechanical properties arising from their complex and heterogeneous atomic environments. However, quantitatively connecting elemental composition to intrinsic strength remains challenging. We develop a composition-dependent stochastic Peierls–Nabarro model that incorporates local compositional fluctuations induced by random site occupancy. Long-range elastic energy is determined by globally averaged material properties, whereas short-range misfit energy depends on local composition. The resulting local Peierls stresses define a statistically constrained optimization problem accounting for both mean lattice resistance and spatial variation. Global elemental fractions are obtained from the optimized local composition field, enabling the prediction of strength-optimized compositions. Numerical results for representative FCC and BCC alloys are compared with experimental data and further used to identify favorable compositions.
MS39 PINNs and AI Agents for Engineering Applications Main organizer: Hyunseok Oh (GIST) · 5 talks
Organizers
Main organizer: Hyunseok Oh (GIST)
- Co-organizer: Jae-Hyuk Lim (Kyunghee University)
- Co-organizer: Seungchul Lee (KAIST)
Program & Speakers
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Geometry-Aware Spatially-Adaptive Operator Learning for Accelerated PDE Simulation
Seungchul Lee (Korea Advanced Institute of Science and Technology)
Operator learning accelerates PDE simulations, but handling geometry variations remains challenging due to localized boundary phenomena that reshape solution fields in spatially non-uniform ways. Existing geometry-aware operators encode shape into a single global descriptor injected at one trunk layer, limiting their ability to capture spatially varying geometric effects near boundaries. We propose GASA-DeepONet, which embeds geometric information spatially and at every trunk layer. A convolutional encoder-decoder produces a spatially resolved basis over the output grid, while a SPADE-inspired module applies position-dependent modulation at each layer. Validated on acoustic wave scattering and jet-impingement cooling, GASA-DeepONet consistently outperforms existing baselines, especially near boundaries with sharp gradients. Combined with CMA-ES, it also enables efficient reliability-based shape optimization under geometric uncertainty.
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Sparse Sensing-Guided Physics-Informed Operator for Real-Time Real-to-Sim Visualization of High-Frequency Vibration Fields
Jae Lim (Kyung Hee University)
This study proposes a sparse sensing-guided physics-informed operator for real-time Real-to-Sim visualization of high-frequency vibration fields from sparse sensor measurements. The proposed framework learns a sensor-to-field operator that maps limited physical sensor signals to simulation-like full-field vibration responses. Ordered sensor histories sampled on a uniform temporal grid are used as an implicit representation of time, enabling full-field reconstruction without explicitly inputting the physical time coordinate. During training, fixed-time-increment numerical differentiation is used to construct physics residuals and impose temporal consistency. To enhance the representation of high-frequency responses, Fourier feature embedding and spectral proper orthogonal decomposition (SPOD)-based modal features are incorporated. The method is evaluated in terms of reconstruction accuracy, phase consistency, noise robustness, and extrapolation stability.
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A Physics-Informed Machine Learning Framework for DIC-Based Physical Field Reconstruction with Robust Data Loss Strategies
Jeong-Hoon Park (Jeonbuk National University)
This study proposes a physics-informed machine learning framework for mechanically consistent reconstruction of noisy and partially reliable DIC-based displacement fields. The framework reconstructs full-field displacements from spatial coordinates and selected DIC measurements while enforcing quasi-static linear elasticity and boundary conditions. Robust data-loss strategies are introduced to address large residuals and spatially nonuniform measurement reliability. Strain and stress fields are subsequently derived using automatic differentiation and the linear elastic constitutive relation. The method was evaluated using two noisy numerical DIC-like examples and an experimental crack-growth DIC dataset. The numerical examples showed good agreement with clean finite element references despite 10% Gaussian noise, while the experimental application demonstrated that the framework can regularize noisy measured displacement fields and recover mechanically plausible derivative fields.
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Physics-Informed Neural Network for Journal Bearing Defect Severity Quantification Using Temperature Sensor Measurements
Minseok Choi (GIST)
Tilting pad journal bearings are critical components in rotating machinery whose wear progression directly affects system reliability and remaining useful life. Real-time wear quantification is challenging due to limited accessibility, and traditional data-driven approaches yield only scalar severity estimates without physical consistency. This work develops a Verification-Conditioned Physics-Informed Neural Network (VC-PINN) to predict the wear-depth landscape from pad-temperature measurements. The Reynolds lubrication and thermohydrodynamic equations serve as physics constraints, with multi-location pad temperatures as data constraints. The wear depth is parameterized with five thermohydrodynamic variables as learnable outputs, trained via a two-phase verification-and-validation strategy with an output-level bias-correction estimator. The approach achieves consistent accuracy and noise robustness, providing a foundation for digital twin and remaining useful life prediction.
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PHM Foundation Models and Multimodal Diagnostic Agents for Overcoming the Interpretation Barrier of High-Frequency Time-Series Signals
Jong Moon Ha (Ajou University)
In rotating-machinery PHM, high-frequency vibration holds the richest health data, yet foundation models and autonomous agents pretrained on language and vision cannot interpret raw waveforms—a modality gap. We propose four complementary ways to make such signals tractable for these models. First, a diagnostic agent couples multi-level signal-processing tools with multimodal reasoning, autonomously deciding when further analysis is needed and synthesizing retrieval-augmented, physics-grounded evidence into a report. Second, a PHM foundation model pretrained on large-scale time-series enables zero-shot fault classification. Third, an interpretation agent describes signal patterns in natural language as a backbone for the diagnostic agent. Fourth, vibration signals are rendered into images where faults become visually salient, diagnosed via a vision-language model (VLM). Components are validated on benchmarks including CWRU, matching expert-based reliability.