  Solution of the discretized Lippmann-Schwinger equation in the spatial
frequency domain involves the inversion of a linear operator specified by the
scattering potential. To regularize this inevitably ill-conditioned problem, we
propose a machine learning approach: a recurrent neural network with long
short-term memory (LSTM) and with the null space projection of the
Lippmann-Schwinger kernel on the recurrence path. The learning method is
trained using examples of typical scattering potentials and their corresponding
scattered fields. We test the proposed method in two cases: electromagnetic
scattering by dielectric objects, and electron scattering by multiple screened
Coulomb potentials. In both cases the solutions to test examples, disjoint from
the training set, were obtained with fewer iterations and were accurate
compared to linear solvers. We also observed surprising generalization ability:
in the electromagnetic case, an LSTM trained with random arrangements of
dielectric spheres was able to obtain the correct solutions for general
topologically similar objects, such as polygons. This suggests that the LSTM
successfully incorporates the physics of scattering into the inversion
algorithm.
