  We consider the inverse acoustic obstacle problem for sound-soft star-shaped
obstacles in two dimensions wherein the boundary of the obstacle is determined
from measurements of the scattered field at a collection of receivers outside
the object. One of the standard approaches for solving this problem is to
reformulate it as an optimization problem: finding the boundary of the domain
that minimizes the $L^2$ distance between computed values of the scattered
field and the given measurement data. The optimization problem is
computationally challenging since the local set of convexity shrinks with
increasing frequency and results in an increasing number of local minima in the
vicinity of the true solution. In many practical experimental settings, low
frequency measurements are unavailable due to limitations of the experimental
setup or the sensors used for measurement. Thus, obtaining a good initial guess
for the optimization problem plays a vital role in this environment.
  We present a neural network warm-start approach for solving the inverse
scattering problem, where an initial guess for the optimization problem is
obtained using a trained neural network. We demonstrate the effectiveness of
our method with several numerical examples. For high frequency problems, this
approach outperforms traditional iterative methods such as Gauss-Newton
initialized without any prior (i.e., initialized using a unit circle), or
initialized using the solution of a direct method such as the linear sampling
method. The algorithm remains robust to noise in the scattered field
measurements and also converges to the true solution for limited aperture data.
However, the number of training samples required to train the neural network
scales exponentially in frequency and the complexity of the obstacles
considered. We conclude with a discussion of this phenomenon and potential
directions for future research.
