  We propose a deep learning approach for wave propagation in media with
multiscale wave speed, using a second-order linear wave equation model. We use
neural networks to enhance the accuracy of a given inaccurate coarse solver,
which under-resolves a class of multiscale wave media and wave fields of
interest. Our approach involves generating training data by the given
computationally efficient coarse solver and another sufficiently accurate
solver, applied to a class of wave media (described by their wave speed
profiles) and initial wave fields. We find that the trained neural networks can
approximate the nonlinear dependence of the propagation on the wave speed as
long as the causality is appropriately sampled in training data. We combine the
neural-network-enhanced coarse solver with the parareal algorithm and
demonstrate that the coupled approach improves the stability of parareal
algorithms for wave propagation and improves the accuracy of the enhanced
coarse solvers.
