Annals of Mathematical Sciences and Applications

Volume 7 (2022)

Number 1

Solving inverse wave scattering with deep learning

Pages: 23 – 48



Yuwei Fan (Stanford University, Stanford, California, U.S.A.)

Lexing Ying (Stanford University, Stanford, California, U.S.A.)


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.


inverse problems, deep learning, inverse scattering, seismic imaging

2010 Mathematics Subject Classification

65N21, 74J25, 86A22

1fundingThe work of Y.F. and L.Y. is partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program. The work of L.Y. is also partially supported by the National Science Foundation under award DMS-1818449.

Received 29 December 2021

Accepted 12 January 2022

Published 7 April 2022