Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method

TL;DR

This paper introduces a neural network-based solver for elliptic equations with singular sources, decomposing solutions into known singular parts and regular parts to improve efficiency and accuracy.

Contribution

The paper proposes a singularity splitting deep Ritz method that effectively handles singular sources in elliptic problems, combining analytical singularity decomposition with neural network solutions.

Findings

The method efficiently solves 2D and multi-dimensional problems with singular sources.

Numerical experiments demonstrate superior accuracy and efficiency over existing neural network approaches.

The approach shows broad applicability to point, line, and combined sources.

Abstract

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. 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.