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A neural operator framework for solving inverse scattering problems

Victor Chenu, Houssem Haddar, Hadrien Montanelli

Abstract

We present a neural operator framework for solving inverse scattering problems. A neural operator produces a preliminary indicator function for the scatterer, which, after appropriate rescaling, is used as a regularization parameter within the Linear Sampling Method to validate the initial reconstruction. The neural operator is implemented as a DeepONet with a fixed radial-basis-function trunk, while the noise level required for rescaling is estimated using a dedicated neural network. A neural tangent kernel analysis guides the architectural design, reducing the network tuning to a single discretization parameter, adjustable according to the wavelength. Two-dimensional numerical experiments demonstrate the method's effectiveness, with a Python toolbox provided for reproducibility.

A neural operator framework for solving inverse scattering problems

Abstract

We present a neural operator framework for solving inverse scattering problems. A neural operator produces a preliminary indicator function for the scatterer, which, after appropriate rescaling, is used as a regularization parameter within the Linear Sampling Method to validate the initial reconstruction. The neural operator is implemented as a DeepONet with a fixed radial-basis-function trunk, while the noise level required for rescaling is estimated using a dedicated neural network. A neural tangent kernel analysis guides the architectural design, reducing the network tuning to a single discretization parameter, adjustable according to the wavelength. Two-dimensional numerical experiments demonstrate the method's effectiveness, with a Python toolbox provided for reproducibility.
Paper Structure (49 sections, 2 theorems, 73 equations, 11 figures, 5 tables)

This paper contains 49 sections, 2 theorems, 73 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Solving the inverse problem with the LSM
  3. Forward and inverse scattering problems
  4. Forward problem
  5. Inverse problem
  6. LSM with Tikhonov--Morozov regularization
  7. Methodology
  8. Tikhonov--Morozov regularization
  9. Practical LSM setup
  10. Sources and sensors
  11. Solving the forward problem
  12. Noise
  13. Computations
  14. Our hybrid LSM approach
  15. Neural operator indicator function
  16. ...and 34 more sections

Key Result

Theorem 1

The following bounds for the spectrum of $K$ holds: where $\sigma_{\min}(P_{\epsilon})$ and $\sigma_{\max}(P_{\epsilon})$ denote the minimum and maximum singular values of $P_{\epsilon}$.

Figures (11)

  • Figure 1: Example setup for the LSM: the probing domain $\Omega$, the defect $D$, sources $\hat{d}_j$, and sensors $\hat{x}_i$. In the multistatic configuration considered here, multiple incident waves are emitted from different directions $\hat{d}_j$ and the scattered field is measured at multiple receiver locations $\hat{x}_i$, yielding a full matrix of measurements.
  • Figure 1: Schematic architecture of our RBF-DeepONet. The trainable branch net generates coefficients in the fixed trunk net basis.
  • Figure 1: Eigenvalues of $50 \times 50$ noisy far-field matrices for a circular obstacle of radius $0.5$, computed with noise levels $\eta = 0.1$ (blue), $\eta = 0.01$ (orange), $\eta = 0.001$ (green), and $\eta = 0$ (red), where $\eta$ is defined in \ref{['noising']}.
  • Figure 1: Comparison of the DeepONet-based indicator (left column), the corresponding LSM indicator (middle column) and the Morozov LSM (right column) for a kite in full aperture. The noise parameter is set respectively to $\eta = 0.05$ (first row), $\eta = 0.1$ (second row) and $\eta = 0.2$ (fourth row). The DeepONet output seems minimally affected by the noise and the associated LSM indicator provides better contrast compared to Morozov LSM, especially for higher noise levels.
  • Figure 2: Morozov regularizer (left) and corresponding LSM indicator function (right) for $m=n=30$. The Morozov regularization function computed from \ref{['morozov']} is used in the Tikhonov-regularized LSM linear system \ref{['regularized_eq']}. This is the standard LSM workflow, which we aim to improve in this paper.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 1: Eigenvalue bound
  • Proof 1