We propose an end-to-end deep learning framework that comprehensively solves the inverse wave scattering problem across all length scales. Our framework consists of the newly introduced wide-band butterfly network coupled with a simple training procedure that dynamically injects noise during training. While our trained network provides competitive results in classical imaging regimes, most notably it also succeeds in the super-resolution regime where other comparable methods fail. This encompasses both (i) reconstruction of scatterers with sub-wavelength geometric features, and (ii) accurate imaging when two or more scatterers are separated by less than the classical diffraction limit. We demonstrate these properties are retained even in the presence of strong noise and extend to scatterers not previously seen in the training set. In addition, our network is straightforward to train requiring no restarts and has an online runtime that is an order of magnitude faster than optimization-based algorithms. We perform experiments with a variety of wave scattering mediums and we demonstrate that our proposed framework outperforms both classical inversion and competing network architectures that specialize in oscillatory wave scattering data.

We introduce an end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) for approximating the inverse scattering map from wide-band scattering data. This architecture incorporates tools from computational harmonic analysis, such as the butterfly factorization, and traditional multi-scale methods, such as the Cooley-Tukey FFT algorithm, to drastically reduce the number of trainable parameters to match the inherent complexity of the problem. As a result WideBNet is efficient: it requires fewer training points than off-the-shelf architectures, and has stable training dynamics, thus it can rely on standard weight initialization strategies. The architecture automatically adapts to the dimensions of the data with only a few hyper-parameters that the user must specify. WideBNet is able to produce images that are competitive with optimization-based approaches, but at a fraction of the cost, and we also demonstrate numerically that it learns to super-resolve scatterers in the full aperture scattering setup.