  Advection-dominated dynamical systems, characterized by partial differential
equations, are found in applications ranging from weather forecasting to
engineering design where accuracy and robustness are crucial. There has been
significant interest in the use of techniques borrowed from machine learning to
reduce the computational expense and/or improve the accuracy of predictions for
these systems. These rely on the identification of a basis that reduces the
dimensionality of the problem and the subsequent use of time series and
sequential learning methods to forecast the evolution of the reduced state.
Often, however, machine-learned predictions after reduced-basis projection are
plagued by issues of stability stemming from incomplete capture of multiscale
processes as well as due to error growth for long forecast durations. To
address these issues, we have developed a \emph{non-autoregressive} time series
approach for predicting linear reduced-basis time histories of forward models.
In particular, we demonstrate that non-autoregressive counterparts of
sequential learning methods such as long short-term memory (LSTM) considerably
improve the stability of machine-learned reduced-order models. We evaluate our
approach on the inviscid shallow water equations and show that a
non-autoregressive variant of the standard LSTM approach that is bidirectional
in the PCA components obtains the best accuracy for recreating the nonlinear
dynamics of partial observations. Moreover---and critical for many applications
of these surrogates---inference times are reduced by three orders of magnitude
using our approach, compared with both the equation-based Galerkin projection
method and the standard LSTM approach.
