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.

翻译：物理信息神经网络（PINNs）是一类利用深度神经网络求解偏微分方程的强大数值方法，已成功应用于众多不同问题。然而，将该方法应用于涉及奇异性（如点源或几何奇异性）的问题时，由于精确解的正则性有限，所获得的近似解通常精度较低。本文研究了在具有几何奇异性及混合边界条件的多边形域中，求解泊松方程的PINNs方法。我们提出了一种新颖的奇异性增强PINN（SEPINN），通过将解析解的奇异行为（如角点奇异性、混合边界条件及边缘奇异性）显式纳入解空间，并给出了该方法的收敛性分析。我们通过二维和三维空间中的大量数值模拟展示了该方法的有效性，并与现有基于神经网络的方法进行了对比研究。