  Model reduction of high-dimensional dynamical systems alleviates
computational burdens faced in various tasks from design optimization to model
predictive control. One popular model reduction approach is based on projecting
the governing equations onto a subspace spanned by basis functions obtained
from the compression of a dataset of solution snapshots. However, this method
is intrusive since the projection requires access to the system operators.
Further, some systems may require special treatment of nonlinearities to ensure
computational efficiency or additional modeling to preserve stability. In this
work we propose a deep learning-based strategy for nonlinear model reduction
that is inspired by projection-based model reduction where the idea is to
identify some optimal low-dimensional representation and evolve it in time. Our
approach constructs a modular model consisting of a deep convolutional
autoencoder and a modified LSTM network. The deep convolutional autoencoder
returns a low-dimensional representation in terms of coordinates on some
expressive nonlinear data-supporting manifold. The dynamics on this manifold
are then modeled by the modified LSTM network in a computationally efficient
manner. An offline unsupervised training strategy that exploits the model
modularity is also developed. We demonstrate our model on three illustrative
examples each highlighting the model's performance in prediction tasks for
fluid systems with large parameter-variations and its stability in long-term
prediction.
