@Article{CiCP-32-4,
author = {Wilson, P., Joshua and Weizhong, Dai and Bora, Aniruddha and Jacob, C., Boyt},
title = {A New Artificial Neural Network Method for Solving Schrödinger Equations on Unbounded Domains},
journal = {Communications in Computational Physics},
year = {2022},
volume = {32},
number = {4},
pages = {1039--1060},
abstract = {<p style="text-align: justify;">The simulation for particle or soliton propagation based on linear or nonlinear Schrödinger equations on unbounded domains requires the computational domain to be bounded, and therefore, a special boundary treatment such as an absorbing
boundary condition (ABC) or a perfectly matched layer (PML) is needed so that the
reflections of outgoing waves at the boundary can be minimized in order to prevent
the destruction of the simulation. This article presents a new artificial neural network
(ANN) method for solving linear and nonlinear Schrödinger equations on unbounded
domains. In particular, this method randomly selects training points only from the
bounded computational space-time domain, and the loss function involves only the
initial condition and the Schrödinger equation itself in the computational domain without any boundary conditions. Moreover, unlike standard ANN methods that calculate
gradients using expensive automatic differentiation, this method uses accurate finite-difference approximations for the physical gradients in the Schrödinger equation. In
addition, a Metropolis-Hastings algorithm is implemented for preferentially selecting
regions of high loss in the computational domain allowing for the use of fewer training
points in each batch. As such, the present training method uses fewer training points
and less computation time for convergence of the loss function as compared with the
standard ANN methods. This new ANN method is illustrated using three examples.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2022-0135},
url = {https://global-sci.com/article/79490/a-new-artificial-neural-network-method-for-solving-schrodinger-equations-on-unbounded-domains}
}