  A common strategy for the dimensionality reduction of nonlinear partial
differential equations relies on the use of the proper orthogonal decomposition
(POD) to identify a reduced subspace and the Galerkin projection for evolving
dynamics in this reduced space. However, advection-dominated PDEs are
represented poorly by this methodology since the process of truncation discards
important interactions between higher-order modes during time evolution. In
this study, we demonstrate that an encoding using convolutional autoencoders
(CAEs) followed by a reduced-space time evolution by recurrent neural networks
overcomes this limitation effectively. We demonstrate that a truncated system
of only two latent-space dimensions can reproduce a sharp advecting shock
profile for the viscous Burgers equation with very low viscosities, and a
six-dimensional latent space can recreate the evolution of the inviscid shallow
water equations. Additionally, the proposed framework is extended to a
parametric reduced-order model by directly embedding parametric information
into the latent space to detect trends in system evolution. Our results show
that these advection-dominated systems are more amenable to low-dimensional
encoding and time evolution by a CAE and recurrent neural network combination
than the POD Galerkin technique.
