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.

翻译：本文提出了一种基于神经网络的高效求解器，用于求解含奇异源的变系数二阶椭圆方程。该类问题涵盖了点源、线源及点线组合源，具有广泛的实际应用价值。所提方法基于将真实解分解为两部分：一部分是利用拉普拉斯方程基本解析解得到的已知奇异部分，另一部分是满足具有更光滑源项的修正椭圆型偏微分方程的正则部分；随后采用深度Ritz方法求解正则部分。我们建议采用路径跟踪策略来选取用于施加狄利克雷边界条件的惩罚参数。通过在二维及多维空间中针对点源、线源及其组合开展大量数值实验，验证了所提方法的有效性；同时与现有几种基于神经网络的方法进行对比研究，结果表明该方法在处理特定问题类别时具有明显竞争力。此外，我们简要讨论了该方法的误差分析。