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Physics informed neural networks (PINNs) represent a very powerful class ofnumerical solvers for partial differential equations using deep neuralnetworks, and have been successfully applied to many diverse problems. However,when applying the method to problems involving singularity, e.g., point sourcesor geometric singularities, the obtained approximations often have lowaccuracy, due to limited regularity of the exact solution. In this work, weinvestigate PINNs for solving Poisson equations in polygonal domains withgeometric singularities and mixed boundary conditions. We propose a novelsingularity enriched PINN (SEPINN), by explicitly incorporating the singularitybehavior of the analytic solution, e.g., corner singularity, mixed boundarycondition and edge singularities, into the ansatz space, and present aconvergence analysis of the scheme. We present extensive numerical simulationsin two and three-dimensions to illustrate the efficiency of the method, andalso a comparative study with existing neural network based approaches.

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