A New Artificial Neural Network Method for Solving Schrödinger Equations on Unbounded Domains (Communications in Computational Physics, 2022)
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 absorbingboundary condition (ABC) or a perfectly matched layer (PML) is needed so that thereflections of outgoing waves at the boundary can be minimized in order to preventthe destruction of the simulation. This article presents a new artificial neural network(ANN) method for solving linear and nonlinear Schrödinger equations on unboundeddomains. In particular, this method randomly selects training points only from thebounded computational space-time domain, and the loss function involves only theinitial condition and the Schrödinger equation itself in the computational domain without any boundary conditions. Moreover, unlike standard ANN methods that calculategradients using expensive automatic differentiation, this method uses accurate finite-difference approximations for the physical gradients in the Schrödinger equation. Inaddition, a Metropolis-Hastings algorithm is implemented for preferentially selectingregions of high loss in the computational domain allowing for the use of fewer trainingpoints in each batch. As such, the present training method uses fewer training pointsand less computation time for convergence of the loss function as compared with thestandard ANN methods. This new ANN method is illustrated using three examples.