报告人：张晨松 研究员 中国科学院数学与系统科学研究院

报告题目：Learning-based Multilevel Solvers for Large-Scale Linear Systems

时间：2025年5月21日10:30—12:00

地点：数学楼2-1会议室

摘要：

This talk presents our efforts on developing learning-based solvers for large-scale linear systems arising from discretized PDEs. Our approach bridges traditional multilevel solvers with machine learning, automating solver design to enhance efficiency and scalability. The method generalizes across grid sizes, coefficients, and right-hand-side terms, enabling offline training and efficient generalization, with convolutional neural networks (CNNs) serving as the basic computational kernels. It utilizes multilevel hierarchy for rapid convergence and cross-level weight sharing to adapt flexibly to varying grid sizes. The proposed solver achieves speedup over classical geometric multigrid methods for convection-diffusion PDEs in preliminary numerical experiments. We further extend this framework by introducing a Fourier neural network (FNN) to accelerate source influence propagation in Helmholtz equations within heterogeneous media. Supervised experiments demonstrate superior accuracy and efficiency compared to other neural operators, while unsupervised scalability tests reveal significant speedups over other AI solvers, achieving near-optimal convergence for wave numbers up to 2000. Ablation studies validate the effectiveness of the multigrid hierarchy and the novel FNN architecture.

报告人简介：

Chen-Song Zhang, PhD. Graduated from the Applied Mathematics & Scientific Computing program at the University of Maryland, College Park, US; Worked as a postdoctoral fellow at the Penn State University, University Park, US; Currently working at the Academy of Mathematics and Systems Science, CAS, China. Main research interests include numerical analysis, adaptive methods, petroleum reservoir simulation, and complex fluid/flow simulation.

邀请人：王飞 教授