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Time-dependent wave equations represent an important class of partialdifferential equations (PDE) for describing wave propagation phenomena, whichare often formulated over unbounded domains. Given a compactly supportedinitial condition, classical numerical methods reduce such problems to boundeddomains using artificial boundary condition (ABC). In this work, we present amachine-learning method to solve this type of equations as an alternative toABCs. Specifically, the mapping from the initial conditions to the PDE solutionis represented by a neural network, trained using wave packets that areparameterized by their band width and wave numbers. The accuracy is tested forboth the second-order wave equation and the Schrodinger equation, including thenonlinear Schrodinger equation. We examine the accuracy from bothinterpolations and extrapolations. For initial conditions lying in the trainingset, the learned map has good interpolation accuracy, due to the approximationproperty of deep neural networks. The learned map also exhibits some goodextrapolation accuracy. We also demonstrate the effectiveness of the method forproblems in irregular domains. Overall, the proposed method provides aninteresting alternative for finite-time simulation of wave propagation.

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