Quasi-optimal domain decomposition method for neural network-based computation of the time-dependent Schrödinger equation

Abstract

In this paper, we derive and analyze the performance of optimal/quasi-optimal Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for the time-dependent Schrödinger equation when implement with neural network-based Partial Differential Equations (PDE) solvers. Optimal SWR methods, which are based on Dirichlet-to-Neumann operators, are known to have a higher convergence rate than classical or optimized SWR methods. However, they are usually considered prohibitive due to their computational costs with standard PDE solvers. Thanks to Physics Informed Neural Network acceleration within the Schwarz waveform relaxation process and an efficient computation of Dirichlet-to-Neumann transmission operators, we demonstrate that optimal and quasi-optimal SWR methods can be performed almost as efficiently as classical or optimized SWR methods while maintaining a faster convergence rate. We present a few numerical examples to illustrate the performance and convergence of the proposed method.