Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks

TL;DR

This work tackles the wide-band inverse scattering problem for the Helmholtz equation by learning an inverse map from far-field data $\{\Lambda^{\omega}\}_{\omega\in\bar{\Omega}}$ to the perturbation $\eta$, rather than solving a nonconvex optimization each time. The authors design an equivariant neural network that encodes translational and rotational symmetries inspired by the filtered back-projection formula, and they further accelerate computation with a butterfly-factorized kernel to compress the back-scattering operator. They present two models: an uncompressed version with built-in equivariance and a compressed version that uses butterfly factorization and a SwitchResNet to maintain nonlinearity in a critical layer; both outperform classical FWI in accuracy at substantially lower cost, with the compressed model achieving further parameter reductions at some loss of sub-Nyquist super-resolution. Experiments on diverse 2D media demonstrate good stability under wide-band data, blazingly fast inference, and competitive performance compared to state-of-the-art ML baselines, highlighting practical scalability for wide-band imaging tasks.

Abstract

This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of classical methods. The architecture is motivated by the filtered back-projection formula in the full aperture regime and with homogeneous background, and it leverages the underlying equivariance of the problem and compressibility of the integral operator. This drastically reduces the number of training parameters, and therefore the computational and sample complexity of the method. In particular, we obtain an architecture whose number of parameters scale sub-linearly with respect to the dimension of the inputs, while its inference complexity scales super-linearly but with very small constants. We provide several numerical tests that show that the current approach results in better reconstruction than optimization-based techniques such as full-waveform inversion, but at a fraction of the cost while being competitive with state-of-the-art machine learning methods.

Figures (10)

• Figure 1: The setup for the inverse scattering problem. In the illustration, the media $\eta$, in the domain of interest $\Omega$, is impinged by the probing wave with frequency $\omega$ from the direction s. The scattered field $u^{\textbf{s}}(\mathbf{x})$ is sampled on the disk $D$.
• Figure 2: Sketch of a family of partitions of a matrix exhibiting the complementary low-rank property. Each sub-matrix induced by the different partition has the same numerical rank.
• Figure 3: An illustration of the matrix factors in the butterfly factorization. In the illustration, $L=6$, $r=1$, and $N = 64$.
• Figure 4: An illustration of the architecture of the model. The shape of each tensor at each step is labelled on the top. Convolutional NN is used to lift the filtering operator.
• Figure 5: The visualization of the application of the underlying equivariance. In the first step, the data matrix $\sf\Lambda^\omega$ is shifted to generate the other four $\sf\Lambda_{\theta_j}$ for $j=0,1,2,3$. Then, the Implementation II $\mathsf{x}\mapsto\operatorname{diag} [(\mathsf{K}^\omega)^\ast \cdot \mathsf{x}\cdot \mathsf{K}^\omega]$ is applied to all four $\sf\Lambda_{\theta_j}$, each of which outputs a row vector. Finally, they are concatenated to form the intermediate representation $\alpha^\omega$.