Recurrent Localization Networks applied to the Lippmann-Schwinger Equation

TL;DR

This paper introduces a recurrent neural network approach to solve the Lippmann-Schwinger equation, combining physics-based modeling with machine learning for efficient and accurate predictions in materials science.

Contribution

It presents a novel machine learning framework using recurrent CNNs to solve Lippmann-Schwinger equations, enabling physics-informed, efficient, and generalizable solutions.

Findings

Achieves high accuracy in elastic strain predictions

Demonstrates effectiveness on two-phase elastic localization

Potential for broad applications in multiscale materials modeling

Abstract

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.