We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier--Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A $\textit{POD-Galerkin PINN ROM}$ is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by two cases which are the steady flow around a backward step and the unsteady turbulent flow around a surface mounted cubic obstacle.

翻译：我们提出了一个减序模型(ROM),利用物理、知情神经网络(PINNs)的最新发展,解决纳维-斯托克方程式(NSE)的反面问题。在拟议办法中,假定流体动态字段存在模拟数据。然后,通过将POD-Galerkin ROM(POD-Galerkin)应用在流体外光谱矩阵上应用POD,并将NSE(或动荡模拟情况下的修改方程式)的Galerkin投射到POD的降基。然后,通过引入深度神经网络($\textit{POD-Galerkin PINN ROM}),将减少的产出与输入的时间和(或)参数相近。神经网络将物理方程式(POD-Galerkin 降低方程式)纳入其结构,作为损失功能的一部分。使用这一方法,减式模型可以将未知的参数(如物理常数或边界条件)大致相近为未知的参数。拟议的ROM是否适用,通过两种情况展示了稳定的地表层稳定向后流和向下流。