Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C369-C398, August 2024.
Abstract. Physics-informed neural networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN, by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition, and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/hhjc-web/SEPINN.git and in the supplementary materials (M160119_SuppMat.pdf [399KB]).