A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains

TL;DR

This paper introduces a neural network-based approach to solve time-dependent wave equations on unbounded domains, offering an alternative to traditional artificial boundary conditions with promising accuracy for various PDEs.

Contribution

The paper presents a novel machine-learning method that maps initial conditions to PDE solutions, effectively handling unbounded domains without artificial boundary conditions.

Findings

Good interpolation accuracy for training initial conditions

Some extrapolation accuracy observed

Effective in irregular domains

Abstract

Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition, classical numerical methods reduce such problems to bounded domains using artificial boundary condition (ABC). In this work, we present a machine-learning method to solve this type of equations as an alternative to ABCs. Specifically, the mapping from the initial conditions to the PDE solution is represented by a neural network, trained using wave packets that are parameterized by their band width and wave numbers. The accuracy is tested for both the second-order wave equation and the Schrodinger equation, including the nonlinear Schrodinger equation. We examine the accuracy from both interpolations and extrapolations. For initial conditions lying in the training set, the learned map has good interpolation accuracy, due to the approximation property of deep neural networks. The learned map also exhibits some good extrapolation accuracy. We also demonstrate the effectiveness of the method for problems in irregular domains. Overall, the proposed method provides an interesting alternative for finite-time simulation of wave propagation.