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

Physics informed neural networks (PINNs) represent a very powerful class of numerical solvers for partial differential equations using deep neural networks, and have been successfully applied to many diverse problems. However, when applying the method to problems involving singularity, e.g., point sources or geometric singularities, the obtained approximations often have low accuracy, due to 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 existing neural network based approaches.