Hu, T; Jin, B; Zhou, Z; (2023) Solving Elliptic Problems with Singular Sources Using Singularity Splitting Deep Ritz Method. SIAM Journal on Scientific Computing , 45 (4) A2043-A2074. 10.1137/22M1520840.

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Abstract

In this work, we develop an efficient solver based on neural networks for secondorder elliptic equations with variable coefficients and a singular source. This class of problems covers general point sources, line sources, and the combination of point-line sources and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two-and multi-dimensional spaces with point sources, line sources, or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.

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