  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, e.g., any
advection-dominated flow phenomena, such as in traffic flow, atmospheric flows,
and air flow over vehicles, a low-dimensional linear subspace poorly
approximates the solution. To address cases such as these, we have developed a
fast and accurate physics-informed neural network ROM, namely 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. 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 1D and 2D Burgers' equations. A speedup of up to
2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are
achieved with an appropriate treatment of the nonlinear terms through a
hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs
are derived that take account of the hyper-reduced operators.
