# Point Source Identification Using Singularity Enriched Neural Networks

arXiv: https://arxiv.org/abs/2408.09143 · HTML: https://mlsanity.com/paper/2408.09143/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 43 references that are in the map.
- First posted on arXiv 17 Aug 2024.

## Builds on (most cited first, 7 of 7)

- [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)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Imaging Conductivity from Current Density Magnitude using Neural Networks](https://mlsanity.com/paper/2204.02441/) (2022)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)

## Built on by (most cited first, 1 of 1)

- [A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems](https://mlsanity.com/paper/2609.27325/) (2026)

## Abstract (arXiv)

The inverse problem of recovering point sources represents an important class of applied inverse problems. However, there is still a lack of neural network-based methods for point source identification, mainly due to the inherent solution singularity. In this work, we develop a novel algorithm to identify point sources, utilizing a neural network combined with a singularity enrichment technique. We employ the fundamental solution and neural networks to represent the singular and regular parts, respectively, and then minimize an empirical loss involving the intensities and locations of the unknown point sources, as well as the parameters of the neural network. Moreover, by combining the conditional stability argument of the inverse problem with the generalization error of the empirical loss, we conduct a rigorous error analysis of the algorithm. We demonstrate the effectiveness of the method with several challenging experiments.