  Reduced-order models are indispensable for multi-query or real-time problems.
However, there are still many challenges to constructing efficient ROMs for
time-dependent parametrized problems. Using a linear reduced space is
inefficient for time-dependent nonlinear problems, especially for
transport-dominated problems. The non-linearity usually needs to be addressed
by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on
the assumption of affine dependence of parameters. This paper proposes and
studies a non-intrusive reduced-order modeling approach for time-dependent
parametrized problems. It is purely data-driven and naturally split into
offline and online stages. During the offline stage, a convolutional
autoencoder, consisting of an encoder and a decoder, is trained to perform
dimensionality reduction. The encoder compresses the full-order solution
snapshots to a nonlinear manifold or a low-dimensional reduced/latent space.
The decoder allows the recovery of the full-order solution from the latent
space. To deal with the time-dependent problems, a high-order dynamic mode
decomposition (HODMD) is utilized to model the trajectories in the latent space
for each parameter. During the online stage, the HODMD models are first
utilized to obtain the latent variables at a new time, then interpolation
techniques are adopted to recover the latent variables at a new parameter
value, and the full-order solution is recovered by the decoder. Some numerical
tests are conducted to show that the approach can be used to predict the unseen
full-order solution at new times and parameter values fast and accurately,
including transport-dominated problems.
