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

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.