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Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problem

Yunwen Yin, Liang Yan

TL;DR

This work proposes a deep decomposition method (DDM) for such purposes, which accomplishes this by providing physical operators associated with the scattering model to the neural network architecture and is the first physics-aware machine learning technique that can have interpretability property for the obstacle detection.

Abstract

In this paper, we consider a deep learning approach to the limited aperture inverse obstacle scattering problem. It is well known that traditional deep learning relies solely on data, which may limit its performance for the inverse problem when only indirect observation data and a physical model are available. A fundamental question arises in light of these limitations: is it possible to enable deep learning to work on inverse problems without labeled data and to be aware of what it is learning? This work proposes a deep decomposition method (DDM) for such purposes, which does not require ground truth labels. It accomplishes this by providing physical operators associated with the scattering model to the neural network architecture. Additionally, a deep learning based data completion scheme is implemented in DDM to prevent distorting the solution of the inverse problem for limited aperture data. Furthermore, apart from addressing the ill-posedness imposed by the inverse problem itself, DDM is the first physics-aware machine learning technique that can have interpretability property for the obstacle detection. The convergence result of DDM is theoretically investigated. We also prove that adding small noise to the input limited aperture data can introduce additional regularization terms and effectively improve the smoothness of the learned inverse operator. Numerical experiments are presented to demonstrate the validity of the proposed DDM even when the incident and observation apertures are extremely limited.

Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problem

TL;DR

This work proposes a deep decomposition method (DDM) for such purposes, which accomplishes this by providing physical operators associated with the scattering model to the neural network architecture and is the first physics-aware machine learning technique that can have interpretability property for the obstacle detection.

Abstract

In this paper, we consider a deep learning approach to the limited aperture inverse obstacle scattering problem. It is well known that traditional deep learning relies solely on data, which may limit its performance for the inverse problem when only indirect observation data and a physical model are available. A fundamental question arises in light of these limitations: is it possible to enable deep learning to work on inverse problems without labeled data and to be aware of what it is learning? This work proposes a deep decomposition method (DDM) for such purposes, which does not require ground truth labels. It accomplishes this by providing physical operators associated with the scattering model to the neural network architecture. Additionally, a deep learning based data completion scheme is implemented in DDM to prevent distorting the solution of the inverse problem for limited aperture data. Furthermore, apart from addressing the ill-posedness imposed by the inverse problem itself, DDM is the first physics-aware machine learning technique that can have interpretability property for the obstacle detection. The convergence result of DDM is theoretically investigated. We also prove that adding small noise to the input limited aperture data can introduce additional regularization terms and effectively improve the smoothness of the learned inverse operator. Numerical experiments are presented to demonstrate the validity of the proposed DDM even when the incident and observation apertures are extremely limited.
Paper Structure (14 sections, 5 theorems, 81 equations, 19 figures, 4 tables)

This paper contains 14 sections, 5 theorems, 81 equations, 19 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem setup and preliminaries
  3. Deep decomposition method
  4. DDM for the inverse obstacle scattering problem
  5. Convergence analysis
  6. The discretization of DDM
  7. Numerical experiments
  8. Experiment Setup
  9. Numerical examples
  10. Example 1: Testing the computational cost and the accuracy
  11. Example 2: Different noise levels
  12. Example 3: Different incident apertures
  13. Example 4: Full aperture case
  14. Summary

Key Result

Lemma 3.1

\newlabellem:opt_lambda For each $\alpha\in \mathbb{R}_{+}$, there exists an optimal curve $\Lambda_{\theta_{BR}^{\ast}}\in U$.

Figures (19)

  • Figure 2.1: \newlabelfig:scattering_modelA schematic illustration of the limited aperture inverse obstacle scattering problem.
  • Figure 3.1: \newlabelfig:DDM_modelA schematic illustration of DDM.
  • Figure 4.1: \newlabelfig:DDM_numerical_modelA schematic illustration of the network structure of DDM in all the numerical examples.
  • Figure 4.2: Example 1: The evolution of the DDM loss function $\mathcal{L}_{DDM}$, the physics-based loss function $\mathcal{L}_{phy}$, and the training relative error $\mathrm{Err}$, throughout the training process.
  • Figure 4.3: Example 1: The real parts of the exact (top) and recovered far-field data (bottom). The x-axis label $i$ represents the $i$-th incident direction, while the y-axis label $j$ indicates the $j$-th observation direction.
  • ...and 14 more figures

Theorems & Definitions (13)

  • Remark 3.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 3 more