Speaker: He Qiaolin, Professor and Doctoral Supervisor, Sichuan University

Date: April 28, 2026

Time: 15:00-16:00 pm

Location: Tencent Meeting: 294 989 480

Sponsor: Research Center for Mathematics and Interdisciplinary Sciences, Shandong University

Abstract:

Solving non-homogeneous partial differential equations (PDEs) in complex geometric domains remains a significant challenge in computational science. While Physics-Informed Holomorphic Neural Networks (PIHNNs) excel at solving homogeneous problems by leveraging complex analysis to automatically satisfy the governing equations, their application is fundamentally limited as they cannot directly address non-homogeneous cases. To overcome this limitation, we propose a novel hybrid framework: the Physics-informed Holomorphic Neural Network combined with traditional numerical methods (HPIHNN). Our approach is founded on the principle of linear superposition, which decomposes the solution into a particular solution and a homogeneous solution. The particular solution, which addresses the non-homogeneous source term, is efficiently computed using spectral methods on a simple, regular, extended domain, thereby circumventing the difficulties associated with meshing complex boundaries. The homogeneous solution is then approximated by a PIHNN, which only needs to be trained on the boundary to satisfy the modified boundary conditions. This strategy preserves the dimensionality-reduction advantage of holomorphic representations and significantly reduces computational cost. We demonstrate the versatility of this framework by applying it to solve non-homogeneous Poisson, Stokes, and biharmonic equations. Numerical experiments on various problems, including those with corner singularities, locking phenomena, and intricate geometries, confirm that PIHNN-S achieves high accuracy and substantially enhances computational efficiency. The synergy between the analytical structure of holomorphic functions, the high-precision of spectral methods, and the flexibility of neural networks provides a powerful and robust paradigm for solving a broad class of challenging physical problems.

For more information, please visit:

https://www.view.sdu.edu.cn/info/1020/211904.htm