  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 equation 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. 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. Therefore, the proposed method provides an interesting alternative
for finite-time simulation of wave propagation.
