Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method
This provides an efficient solver for elliptic problems with singular sources (point/line sources), which has practical applications but represents an incremental improvement over existing neural network approaches.
The authors developed a neural network solver for second-order elliptic equations with singular sources by splitting the solution into singular and regular parts, then solving the regular part with the deep Ritz method. Numerical experiments in 2D and higher dimensions showed the approach is competitive with existing neural network methods for this specific problem class.
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.