Machine Learning Regularized Solution of the Lippmann-Schwinger Equation

Abstract

The Lippmann-Schwinger equation has been applied on various branches of physics, especially optical and quantum scattering. Solving the equation requires the inversion of a linear operator specified by the scattering potential, which is ill-conditioned. To resolve numerical difficulty originating from such ill-conditionedness, we propose a machine learning approach to find an appropriate regularization. Inspired by the proximal algorithm, we try to solve the equation with a hybridization of the physical operator and a regularizing network: a recurrent neural network with long short-term memory (LSTM).

We train the LSTM using typical scattering potentials and their corresponding scattered fields. For the evaluation of the LSTM, two scattering cases are considered: electromagnetic scattering by dielectric objects, and electron scattering by multiple screened Coulomb potentials. It is observed that the network can estimate scattered fields that are comparable to those from linear solvers with fewer iterations. We also observed surprising generalization ability. Specifically, in the electromagnetic case, the LSTM trained with objects consisting of dielectric spheres can estimate reasonable solutions for general topologically similar objects, such as polygons. This suggests that the scattering physics is properly fused to the network through the training process.