POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier-Stokes equations

TL;DR

This paper introduces a novel reduced order modeling approach combining POD-Galerkin methods with physics-informed neural networks to efficiently solve inverse problems in fluid dynamics governed by Navier-Stokes equations.

Contribution

It develops a POD-Galerkin PINN ROM that integrates physical laws into neural networks for inverse problems in fluid flow modeling.

Findings

Successfully applied to steady backward step flow.

Effectively estimated unknown parameters in turbulent flow.

Demonstrated accuracy and efficiency in complex flow scenarios.

Abstract

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.