Special Session 151: Encounter and Merging of Mesh-based Methods and Meshless Methods in the Era of Machine Learning

Massively parallel domain decomposition methods for solving the Helmholtz equations in $\mathcal{O}(k)$ runtime

Shihua Gong The Chinese University of Hong Kong, Shenzhen, SICIAM and SLAI Peoples Rep of China Co-Author(s):

Abstract: We present a novel massively parallel variant of the Restricted Additive Schwarz method with Perfectly Matched Layers (RAS-PML) as a fixed point iteration for solving the Helmholtz equation in $d$-dimensional space and at high-frequency $k$. We are able to solve $\mathcal{O}(k^d)$-scale linear systems coming from $\mathcal{O}(k^{-1})$-diameter mesh discretizations of the Helmholtz problems in $\mathcal{O}(k)$ runtime with $\mathcal{O}(k^d)$ processors. The method is motivated by the theory in {\tt arXiv:2404.02156}, where the authors proved that a related Schwarz method using fixed subdomain and PML configuration has a contraction rate which decays super-algebraically fast with respective to $k\to \infty$. The proposed algorithm can be implemented easily using standard routines in the parallel domain decomposition packages.

A penalty approach for topology optimization problems

Wei Gong Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Yuanda Ye

Abstract: We propose a novel penalty method framework for compliant mechanism problems, incorporating a convex nonlocal perimeter approximation scheme. 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, ensuring consistency across scales. 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, with results supported by convergence analysis.

Transformer: structure conforming operator learning

Ruchi Guo Sichuan University Peoples Rep of China Co-Author(s):    Shuhao Cao, Long Chen, and Ruchi Guo

Abstract: The Transformer has emerged as one of the most advanced neural network architectures, with wide applications in large language models (LLMs), AI for Science, and image/video process. Despite its success, its mathematical foundations remain largely open. This research presents our recent progress toward addressing this gap, structured in two parts. First, we introduce a new perspective based on Petrov-Galerkin projection and Fourier analysis to better interpret the attention mechanism. Building on this framework, we propose a modified Transformer architecture that admits a clearer mathematical interpretation and exhibits a frequency-bootstrapping property. Second, drawing inspiration from direct sampling methods (DSMs) for inverse problems, we develop a novel feature generation approach: data features are constructed by solving PDEs and then incorporated into the attention mechanism. By embedding a learnable nonlocal kernel, the DSM is naturally reformulated as such the modified attention mechanism. We demonstrate the proposed method on electrical impedance tomography (EIT), a prototypical severely ill-posed nonlinear inverse problem, which achieves superior accuracy over its predecessors and contemporary operator learners.

Lattice-Based Modeling of Cell Migration Dynamics

Yucheng Hu CNU Peoples Rep of China Co-Author(s):

Abstract: Cell migration is a fundamental process in many biological phenomena. In this talk, I will present a lattice-based framework for modeling the dynamics of migrating cells. The lattice formulation provides a simple and intuitive way to study how microscopic movement rules give rise to collective behaviors at larger scales. Depending on the modeling assumptions, the framework can capture important features such as random motility, directional bias, adhesion. The emphasis will be on the modeling ideas and on the role of regular lattice structures as a useful tool for understanding cell migration dynamics.

On the interplay between iterative methods and error estimators

Yuwen Li Zhejiang University Peoples Rep of China Co-Author(s):    Han Shui

Abstract: This talk presents a family of a posteriori error estimates motivated by iterative solvers. We use simple smoothers (one step of Jacobi/Gauss-Seidel iteration), on an auxiliary finer mesh to process the finite element residual for a posteriori error control. The proposed error estimators are user-friendly and robust with respect to polynomial degrees and physical parameters. Numerical experiments include Biot, convection-diffusion, Helmholtz, and Maxwell equations. The material is based on joint works with Ludmil Zikatanov and Han Shui.

Computational Imaging with Generative Models

Jiulong Liu Academy of Mathematics and Systems Science,CAS Peoples Rep of China Co-Author(s):

Abstract: Bayesian statistical inversion and sparsity-based methods have long been effective mathematical tools for reducing sampling requirements in compressive sensing, enabling applications in underdetermined imaging systems such as MRI and CT. With the rapid advancement of deep learning, a new class of methods has emerged that learns data-driven representations, offering enhanced performance in signal and image reconstruction tasks. To address underdetermined and ill-conditioned inverse problems with limited measurements, we develop compressive sensing and Bayesian reconstruction frameworks that incorporate generative model-based priors. These approaches achieve improved reconstruction quality and computational efficiency compared to traditional regularization techniques and existing data-driven methods. Moreover, we establish theoretical guarantees for recovery performance under these generative priors. In this talk, I will present these methods and highlight recent results in applications including MRI reconstruction, phase retrieval, and other nonlinear inverse problems.

Structure-preserving parametric finite element method for surface diffusion

Chunmei Su Tsinghua University Peoples Rep of China Co-Author(s):

Abstract: We propose a novel formulation for parametric finite element methods to simulate surface diffusion, which incorporates two scalar Lagrange multipliers and two evolution equations involving the surface area and volume, respectively, to ensure that the resulting numerical methods preserve the geometric structure of surface diffusion, i.e., the area-decreasing and volume-preserving properties. By discretizing the spatial variable using linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop several high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton`s method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of surface diffusion.

Machine learning method with numerical Integration for singular problems

Hehu Xie Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Zhongshuo Lin and Yifan Wang

Abstract: We present MRpro, an open-source image reconstruction package for MRI built upon PyTorch and open data formats. MRpro consists of three main areas. First, it provides unified datastructures for the processing and handling of MR datasets and their associated metadata, such as sampling trajectories in the Fourier domain. Second, it provides relevant mathematical concepts and operations required to build abstract reconstruction algorithms. Third, to enable the easy development of state-of-the-art learned reconstruction methods, MRpro includes essential building blocks such as so-called data-consistency layers, differentiable optimization layers and well-established neural network backbones. In this talk, we showcase different components of MRpro by considering several learned and non-learned MR imaging problems using both publicly available open-source data, numerical phantoms as well as raw scanner data, demonstrating MRpro`s versatility.

Quasi-entropy: in free energies and closure approximations

Jie Xu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Jie Xu

Abstract: 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.

A morphology-adaptive random feature method for inverse source problem of the Helmholtz equation

Haijun Yu Academy of Mathematics and Systems Science Peoples Rep of China Co-Author(s):    Xinwei Hu, Jingrun Chen and Haijun Yu

Abstract: We propose a Morphology-Adaptive Random Feature Method (MARFM), a novel two-phase framework that adaptively locates critical regions and adds morphology activation functions for tackling the multi-frequency inverse source problem for the Helmholtz equation with complex geometry. Our framework recasts the ill-posed inverse problem into a well-posed, strictly convex optimization problem by reformulating the governing Helmholtz equation as a Tikhonov-regularized integral equation via its fundamental solution. In the first stage, the Integral Adaptive RFM (IA-RFM), employs an adaptive algorithm to rapidly localize the source support, thereby reducing computational overhead and accelerating convergence. In the second stage, posterior geometric information is progressively integrated into the solver via hybrid basis functions, enabling a precise reconstruction of complex morphologies. The MA-RFM extends the capabilities of RFM to handle PDEs with singular solutions while preserving its mesh-free efficiency. We demonstrate the superior performance of our approach through ample challenging 2D and 3D benchmark problems, even under limited and noisy measurement conditions, highlighting its robustness and accuracy in reconstructing complex and disjoint sources.

Some trainer friendly meshless methods

Shuo Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: The training process for the neural network problems may suffer from the super parameters and the non-convex formulation. Typical examples include the boundary condition imposition by penalty and the problem of saddle-point essence. This talk discusses how to impose the boundary condition for PINN and how to solve the saddle-point formulation physical model in friendly ways. The main proposal is to formulate the problems as convex energy-minimization problems to be friendly for optimizers. Though neural network problems are used for illustration, the methods essentially work for general meshless methods.

Weak Generative Sampler

Xiang ZHOU City University of Hong Kong Hong Kong Co-Author(s):    Zhiqiang Cai, Yu Cao, Yuanfei Huang,

Abstract: The current deep learning-based method solves the stationary Fokker--Planck equation to determine the invariant probability density function in the form of deep neural networks, but they generally do not directly address the problem of sampling from the computed density function. Traditional numerical solvers for stochastic differential equations require both a fine step size and a lengthy simulation period, resulting in biased and correlated samples. In this work, we introduce a framework that employs a weak generative sampler (WGS) to directly generate independent and identically distributed (iid) samples induced by a transformation map derived from the stationary Fokker--Planck equation. Our proposed loss function is based on the weak form of the Fokker--Planck equation, integrating normalizing flows to characterize the invariant distribution and facilitate sample generation from a base distribution. Our randomized test function circumvents the need for min-max optimization in the traditional weak formulation. Our method necessitates neither the computationally intensive calculation of the Jacobian determinant nor the invertibility of the transformation map. A crucial component of our framework is the adaptively chosen family of test functions in the form of Gaussian kernel functions with centers related to the generated data samples. Experimental results on several benchmark examples demonstrate the effectiveness and scalability of our method, which offers both low computational costs and excellent capability in exploring multiple metastable states.

Efficient simulation under a population genetics model of carcinogenesis

Tianqi Zhu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: Cancer results from somatic mutations disrupting cell division, with the number of required mutations varying by cancer type. The waiting time for m mutations in a cell is a key parameter in carcinogenesis models, which are theoretically complex due to interactions of mutation, drift, and selection, and computationally expensive to simulate due to large cell populations and low mutation rates. We present an efficient algorithm to simulate the waiting time under a population genetics model of cancer. Our hybrid method combines an exact algorithm for small populations with a coarse-grained $\tau$-leaping approximation for large populations. Comparisons with exact simulations for small populations and asymptotic results for large populations confirm the algorithm`s accuracy and computational efficiency. We applied it to study waiting times for up to 20 mutations in a Moran model with variable population sizes. This algorithm may facilitate the study of realistic carcinogenesis models incorporating variable mutation rates and fitness effects.