  The bulk of computational approaches for modeling physical systems in
materials science derive from either analytical (i.e. physics based) or
data-driven (i.e. machine-learning based) origins. In order to combine the
strengths of these two approaches, we advance a novel machine learning approach
for solving equations of the generalized Lippmann-Schwinger (L-S) type. In this
paradigm, a given problem is converted into an equivalent L-S equation and
solved as an optimization problem, where the optimization procedure is
calibrated to the problem at hand. As part of a learning-based loop unrolling,
we use a recurrent convolutional neural network to iteratively solve the
governing equations for a field of interest. This architecture leverages the
generalizability and computational efficiency of machine learning approaches,
but also permits a physics-based interpretation. We demonstrate our learning
approach on the two-phase elastic localization problem, where it achieves
excellent accuracy on the predictions of the local (i.e., voxel-level) elastic
strains. Since numerous governing equations can be converted into an equivalent
L-S form, the proposed architecture has potential applications across a range
of multiscale materials phenomena.
