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

TL;DR

This paper introduces a novel singularity enriched PINN method for solving Poisson problems in polygonal domains, explicitly modeling singularities to improve accuracy near geometric irregularities.

Contribution

The work proposes SEPINN, which incorporates singularity behavior into neural network solutions for better handling of geometric singularities in Poisson problems.

Findings

SEPINN achieves higher accuracy near singularities.

Numerical simulations demonstrate improved convergence.

Comparative studies show advantages over existing methods.

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 (SEPINN), 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.