Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Point Source Identification Using Singularity Enriched Neural Networks

Tianhao Hu, Bangti Jin, Zhi Zhou

TL;DR

This paper tackles the inverse source problem for point sources in the Poisson equation by introducing Singularity Enriched Neural Networks (SENN), which separate the singular part of the solution (captured by the fundamental solution) from a regular part approximated by a neural network. It develops a two-stage approach: first detecting the number of sources M via a reciprocity-gap/Hankel framework, then reconstructing intensities and locations through a nonconvex empirical loss that couples the NN-based regular part with the singular sources. The authors provide Hölder conditional stability bounds and a generalization-error analysis grounded in Rademacher complexity, yielding explicit error estimates for source locations, densities, and the recovered state in terms of noise, network architecture, and sampling. Numerical experiments across Poisson, Helmholtz, and high-dimensional settings demonstrate accurate, robust recovery under noise, highlighting SENN’s effectiveness and potential for extensions to more general source geometries and time-dependent problems.

Abstract

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.

Point Source Identification Using Singularity Enriched Neural Networks

TL;DR

This paper tackles the inverse source problem for point sources in the Poisson equation by introducing Singularity Enriched Neural Networks (SENN), which separate the singular part of the solution (captured by the fundamental solution) from a regular part approximated by a neural network. It develops a two-stage approach: first detecting the number of sources M via a reciprocity-gap/Hankel framework, then reconstructing intensities and locations through a nonconvex empirical loss that couples the NN-based regular part with the singular sources. The authors provide Hölder conditional stability bounds and a generalization-error analysis grounded in Rademacher complexity, yielding explicit error estimates for source locations, densities, and the recovered state in terms of noise, network architecture, and sampling. Numerical experiments across Poisson, Helmholtz, and high-dimensional settings demonstrate accurate, robust recovery under noise, highlighting SENN’s effectiveness and potential for extensions to more general source geometries and time-dependent problems.

Abstract

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.
Paper Structure (11 sections, 15 theorems, 79 equations, 5 figures, 6 tables, 1 algorithm)

This paper contains 11 sections, 15 theorems, 79 equations, 5 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The proposed method
  3. Detection of the number of sources
  4. Singularity enrichment
  5. Reconstruction via SENN
  6. Error analysis
  7. Conditional stability
  8. Generalization error analysis of the empirical loss
  9. Numerical experiments and discussions
  10. Conclusions
  11. Technical estimates

Key Result

Proposition 2.1

Let the Hankel matrix $A=(a_{ij})_{\overline{M}\times\overline{M}}$, with $a_{ij}=\mathcal{R}(w_{i+j-2})\in\mathbb{C}$. Then ${\rm rank}(A)=M$ and the first $M$ columns of $A$ are linearly independent.

Figures (5)

  • Figure 1: The neural network approximation of $u(\mathbf{x})$ for Example \ref{['exam:poisson']}, case (i).
  • Figure 2: The dynamics of the training process for the proposed method for Example \ref{['exam:poisson']}, case (i): (a) the decay of the loss $\widehat{L}^\delta$ versus the iteration index $i$, (b) the error $e$ versus the iteration index $i$.
  • Figure 3: The training dynamics of the density $c$ and location $\mathbf{x}=(x_1,x_2)$ of the four point sources for Example \ref{['exam:poisson']}, case (i).
  • Figure 4: The NN approximation of the real (top) and imaginary (bottom) parts of $u(\mathbf{x})$ for Example \ref{['exam:hel']}, case (i), at the slice $x_3=0$.
  • Figure 5: The dynamics of the training process for the proposed method for Example \ref{['exam:hel']}, case (i): (a) the decay of the loss $\widehat{L}^\delta$ versus the iteration index $i$; (b) the training dynamics of one singular point (i.e. the intensity $c$ and the locations) ; (c) the relative error $e$ versus the iteration index $i$.

Theorems & Definitions (30)

  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3: Hall-Rado Rado:1949
  • proof : Proof of Theorem \ref{['thm:disbound-1']}
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 20 more