  Rapid simulations of advection-dominated problems are vital for multiple
engineering and geophysical applications. In this paper, we present a long
short-term memory neural network to approximate the nonlinear component of the
reduced-order model (ROM) of an advection-dominated partial differential
equation. This is motivated by the fact that the nonlinear term is the most
expensive component of a successful ROM. For our approach, we utilize a
Galerkin projection to isolate the linear and the transient components of the
dynamical system and then use discrete empirical interpolation to generate
training data for supervised learning. We note that the numerical
time-advancement and linear-term computation of the system ensure a greater
preservation of physics than does a process that is fully modeled. Our results
show that the proposed framework recovers transient dynamics accurately without
nonlinear term computations in full-order space and represents a cost-effective
alternative to solely equation-based ROMs.
