Learning solutions of parametric Navier-Stokes with physics-informed neural networks

TL;DR

This paper introduces a physics-informed neural network approach to solve parametric Navier-Stokes equations, enabling accurate flow predictions across different parameters by incorporating PDE constraints into the training process.

Contribution

The paper presents a novel PINN-based method that effectively learns parametric solutions of Navier-Stokes equations, overcoming convergence issues of traditional PINNs in nonlinear PDEs.

Findings

PINNs accurately predict flow velocities and pressure over Reynolds number range.

PINNs outperform conventional neural networks in gradient and vorticity predictions.

The approach ensures physical law compliance in flow simulations.

Abstract

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.