  This paper is concerned with the inverse problem of reconstructing an
inhomogeneous medium from the acoustic far-field data at a fixed frequency in
two dimensions. This inverse problem is severely ill-posed (and also strongly
nonlinear), and certain regularization strategy is thus needed. However, it is
difficult to select an appropriate regularization strategy which should enforce
some a priori information of the unknown scatterer. To address this issue, we
plan to use a deep learning approach to learn some a priori information of the
unknown scatterer from certain ground truth data, which is then combined with a
traditional iteration method to solve the inverse problem. Specifically, we
propose a deep learning-based iterative reconstruction algorithm for the
inverse problem, based on a repeated application of a deep neural network and
the iteratively regularized Gauss-Newton method (IRGNM). Our deep neural
network (called the learned projector in this paper) mainly focuses on learning
the a priori information of the shape of the unknown contrast with a
normalization technique in the training process and is trained to act like a
projector which is helpful for projecting the solution into some feasible
region. Extensive numerical experiments show that our reconstruction algorithm
provides good reconstruction results even for the high contrast case and has a
satisfactory generalization ability.
