  This paper proposes a neural network approach for solving two classical
problems in the two-dimensional inverse wave scattering: far field pattern
problem and seismic imaging. The mathematical problem of inverse wave
scattering is to recover the scatterer field of a medium based on the boundary
measurement of the scattered wave from the medium, which is high-dimensional
and nonlinear. For the far field pattern problem under the circular
experimental setup, a perturbative analysis shows that the forward map can be
approximated by a vectorized convolution operator in the angular direction.
Motivated by this and filtered back-projection, we propose an effective neural
network architecture for the inverse map using the recently introduced BCR-Net
along with the standard convolution layers. Analogously for the seismic imaging
problem, we propose a similar neural network architecture under the rectangular
domain setup with a depth-dependent background velocity. Numerical results
demonstrate the efficiency of the proposed neural networks.
