  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.
