  It is challenging to construct generalized physical models of wave
propagation in nature owing to their complex physics as well as widely varying
environmental parameters and dynamical scales. In this article, we present the
convolutional autoencoder recurrent network (CRAN) as a data-driven model for
learning wave propagation phenomena. The CRAN consists of a convolutional
autoencoder for learning low-dimensional system representation and a long
short-term memory recurrent neural network for the system evolution in low
dimension. We show that the convolutional autoencoder significantly outperforms
the dimension-reduction of complex wave propagation phenomena via
projection-based methods as it can directly learn subspaces resembling wave
characteristics. On the other hand, the projection-based modes are restricted
to the Fourier subspace. Geometric priors of the convolutional autoencoder
enabling selective scale separation of complex wave dynamics further enhance
its dimension-reduction capability. We also demonstrate that geometric priors
such as translation equivariance and translational invariance of the
convolutional autoencoder enable generalized learning of low-dimensional maps.
Thus, the composite CRAN model connecting the convolutional autoencoder with a
long short-term memory network specially designed for autoregressive modeling
can perform generalized wave propagation prediction over the desired time
horizon. Numerical experiments display 90% mean structural similarity index
measure of CRAN predictions compared to true solutions for out-of-training
cases, and less than 10% pointwise $L_1$ error for most cases, verifying such
generalization claims. Finally, the CRAN predictions offer similar wave
characteristic patterns to the target solutions indicating not only their
generalization but also their kinematical consistency.
