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The learned range test method for the inverse inclusion problem

Shiwei Sun, Giovanni S. Alberti

TL;DR

The paper tackles reconstructing an insulating inclusion $B$ inside a bounded domain $\Omega$ from a single boundary measurement for the Laplace equation. It recasts the range test (RT) into a neural network, introducing the learned range test (LRT), and augments it with a distance-based classifier to handle inclusions at varying distances from the boundary. Through a three-step strategy combining training on distance-similar datasets, a classifier, and a weighted reconstruction, the method yields accurate, stable polygonal-inclusion reconstructions and outperforms both the deterministic RT and end-to-end neural networks, with substantial speedups after training. The approach demonstrates how model-based qualitative methods can be enhanced by data-driven learning, offering a practical, interpretable framework for one-measurement inverse problems and suggesting paths for extension to more complex geometries and multiple measurements.

Abstract

We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $Ω\subset\mathbb{R}^d$ from a single pair of Cauchy data $(u|_{\partialΩ},\partial_νu|_{\partialΩ})$, where $Δu=0$ in $Ω\setminus\overline B$ and $u=0$ on $\partial B$. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.

The learned range test method for the inverse inclusion problem

TL;DR

The paper tackles reconstructing an insulating inclusion inside a bounded domain from a single boundary measurement for the Laplace equation. It recasts the range test (RT) into a neural network, introducing the learned range test (LRT), and augments it with a distance-based classifier to handle inclusions at varying distances from the boundary. Through a three-step strategy combining training on distance-similar datasets, a classifier, and a weighted reconstruction, the method yields accurate, stable polygonal-inclusion reconstructions and outperforms both the deterministic RT and end-to-end neural networks, with substantial speedups after training. The approach demonstrates how model-based qualitative methods can be enhanced by data-driven learning, offering a practical, interpretable framework for one-measurement inverse problems and suggesting paths for extension to more complex geometries and multiple measurements.

Abstract

We consider the inverse problem consisting of the reconstruction of an inclusion contained in a bounded domain from a single pair of Cauchy data , where in and on . We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.

Paper Structure

This paper contains 26 sections, 1 theorem, 35 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The inverse problem
  3. Mathematical formulation
  4. Literature review
  5. Review of the range test
  6. Theoretical analysis of the RT
  7. Numerical implementations of the RT
  8. The learned range test method
  9. The RT as a neural network
  10. The neural network with learned weights
  11. Three-step strategy
  12. Numerical experiments
  13. Setup
  14. Construction of the dataset
  15. Training of the networks
  16. ...and 11 more sections

Key Result

Theorem 3.1

Assume that $B\Subset \Omega$ is a convex polygon satisfying the distance property: where $\mathrm{diam}(B)$ represents the diameter of $B$, and $\mathrm{dist}(B, \partial \Omega)$ is the distance between $B$ and $\partial \Omega$. Assuming that the test domain $G$ is a convex polygon, we have:

Figures (13)

  • Figure 1: (a): Some test domains in $P(x_i)$. In (b) and (c), the blue square represents the inclusion $B$. (b): if $x_i \in B$, then all test domains in $P(x_i)$ are not positive. (c): if $x_i \not\in B$, then there exists a positive test domain in $P(x_i)$, for instance, the red one.
  • Figure 2: The red polygon represents the target $B$. (a) and (c) are the results generated by using noise-free measurement, whereas (b) and (d) correspond to the measurement with $3\%$ noise.
  • Figure 3: The architecture of the NN relative to the RT method.
  • Figure 4: The diagram of the three-step strategy. Here, $y^\delta(B)$ denotes the noisy version of $\partial_\nu\omega(B)$ with noise level $\delta$, see \ref{['eq:def_noise']}.
  • Figure 5: The architecture of the classifier.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 3.1: Lin-2021-15