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.

提出一种基于神经网络减小秩的 ROM 新方法，在处理对流主导的现象时更好地逼近高保真模型，通过在非线性流域中超减少运算并得到适当的误差界限达到更好的效率。