  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.
