Chen Cui, Doctor, Yau Mathematical Sciences Center

Chensong Zhang, Professor

Neural Solvers for Large-Scale Linear Systems of Equations

10:00-11:00 December 26(Friday), S615

Deep learning-based numerical methods for PDEs have rapidly become a focal point in scientific computing, with Neural Solvers for discretized PDE systems emerging as a crucial branch. In this report, we first introduce the Fourier Neural Solver (FNS), a complementary hybrid iterative method based on the Fast Fourier Transform. The FNS exploits the spectral bias of deep learning alongside simple iterative methods known for smoothing effect. We then enhance the FNS's robustness and convergence speed by incorporating transition operators and learnable smoothers based on eigen-decomposition. To further extend the method's applicability to unstructured grids and PDE systems, we develop an improved FNS that integrates Graph Neural Networks with learnable basis transformation techniques. Numerical results on various finite element discretized systems validate the effectiveness of these approaches. Moreover, for the challenging high-wavenumber Helmholtz equation, we present a Wave-ADR Neural Solver. By learning characteristic error near the kernel space, this solver significantly enhances the performance of traditional multigrid methods on high-frequency problems.

报告人简介：崔晨，清华大学丘成桐数学中心博士后研究员。本科毕业于吉林大学，于湘潭大学获得博士学位。他的主要研究领域是科学机器学习，特别是数据驱动的多尺度计算方法，比如深度学习增强的多重网格法、区域分解法、降阶基方法、多尺度有限元以及神经算子预处理等。最近，他关注黎曼优化算法及其在深度学习流形优化中的应用研究。目前，已在包括 SIAM J. Sci. Comput., J. Comput. Phys., J. Comput. Appl. Math. 在内的计算数学权威期刊上发表多篇学术论文。