mlsanityData to 14 Oct 2026

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

arXiv 2209.02931 ↗First posted 7 Sep 2022math.NA cs.LG cs.NAcited by 4

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Abstract (arXiv)

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.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 22 of its 78 references that are in the map. First posted on arXiv 7 Sep 2022.

Builds on

22 of 78 references.

References · 22

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10 more
  1. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 1; 2025: 1; 2026: 1.

4 papers.

  1. 2023Solving Elliptic Optimal Control Problems via Neural Networks and Optimality Systemcited by 3
  2. 2025Learning Singularity-Encoded Green's Functions with Application to Iterative Methodscited by 3
  3. 2024Point Source Identification Using Singularity Enriched Neural Networkscited by 1
  4. 2026A hybrid analytical-PINN model for subsurface simulation of geothermal heat exchangers in heterogeneous undergroundcited by 0