  Traditional linear subspace reduced order models (LS-ROMs) are able to
accelerate physical simulations, in which the intrinsic solution space falls
into a subspace with a small dimension, i.e., the solution space has a small
Kolmogorov n-width. However, for physical phenomena not of this type, such as
advection-dominated flow phenomena, a low-dimensional linear subspace poorly
approximates the solution. To address cases such as these, we have developed an
efficient nonlinear manifold ROM (NM-ROM), which can better approximate
high-fidelity model solutions with a smaller latent space dimension than the
LS-ROMs. Our method takes advantage of the existing numerical methods that are
used to solve the corresponding full order models (FOMs). The efficiency is
achieved by developing a hyper-reduction technique in the context of the
NM-ROM. Numerical results show that neural networks can learn a more efficient
latent space representation on advection-dominated data from 2D Burgers'
equations with a high Reynolds number. A speed-up of up to 11.7 for 2D Burgers'
equations is achieved with an appropriate treatment of the nonlinear terms
through a hyper-reduction technique.
