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：在本研究中，我们基于神经网络开发了一种对具有变系数和奇异源的二阶椭圆方程进行高效求解的求解器。这类问题包括一般点源、线源以及点源和线源组合，具有广泛的实际应用。所提出的方法基于将真实解分解为已知解析奇异分量和满足带有平滑源的适当修改椭圆PDE的常规分量，然后使用深里兹方法求解常规分量。建议采用路径跟踪策略为强制迪利克雷边界条件选择惩罚参数。在二维和多维空间中进行了广泛的数值实验，包括点源、线源或它们的组合，以说明所提出方法的效率，并进行与几种现有基于神经网络的方法的比较研究，结果清楚地表明了其对于特定类别的问题的竞争力。此外，我们简要讨论了该方法的误差分析。