  We consider in this work an inverse acoustic scattering problem when only
phaseless data is available. The inverse problem is highly nonlinear and
ill-posed due to the lack of the phase information. Solving inverse scattering
problems with phaseless data is important in applications as the collection of
physically acceptable phased data is usually difficult and expensive. A novel
direct sampling method (DSM) will be developed to effectively estimate the
locations and geometric shapes of the unknown scatterers from phaseless data
generated by a very limited number of incident waves. With a careful
theoretical analysis of the behavior of the index function and some
representative numerical examples, the new DSM is shown to be computationally
efficient, easy to implement, robust to large noise, and does not require any
prior knowledge of the unknown scatterers. Furthermore, to fully exploit the
index functions obtained from the DSM, we also propose to integrate the DSM
with a deep learning technique (DSM-DL) to achieve high-quality
reconstructions. Several challenging and representative numerical experiments
are carried out to demonstrate the accuracy and robustness of reconstructions
by DSM-DL. The DSM-DL networks trained by phased data are further theoretically
and numerically shown to be able to solve problems with phaseless data.
Additionally, our numerical experiments also show the DSM-DL can solve inverse
scattering problems with mixed types of scatterers, which renders its
applications in many important practical scenarios.
