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

arXiv: https://arxiv.org/abs/2209.02931 · HTML: https://mlsanity.com/paper/2209.02931/

## 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 (most cited first, 22 of 22)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint](https://mlsanity.com/paper/1912.04862/) (2019)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Error Estimates for the Deep Ritz Method with Boundary Penalty](https://mlsanity.com/paper/2103.01007/) (2021)
- [Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.01394/) (2021)
- [Imaging Conductivity from Current Density Magnitude using Neural Networks](https://mlsanity.com/paper/2204.02441/) (2022)
- [A Primal Dual Active Set with Continuation Algorithm for the ℓ0-Regularized Optimization Problem](https://mlsanity.com/paper/1403.0515/) (2014)

## Built on by (most cited first, 4 of 4)

- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)
- [Point Source Identification Using Singularity Enriched Neural Networks](https://mlsanity.com/paper/2408.09143/) (2024)
- [A hybrid analytical-PINN model for subsurface simulation of geothermal heat exchangers in heterogeneous underground](https://mlsanity.com/paper/2607.12271/) (2026)

## 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.