Learning Solutions of Parametric Navier-Stokes With Physics-Informed Neural Networks

17 Pages Posted: 19 Feb 2024

See all articles by Mahdi Naderibeni

Mahdi Naderibeni

affiliation not provided to SSRN

Marcel Reinders

Leiden University - Leiden Computational Biology Center

Liang Wu

affiliation not provided to SSRN

David M.J. Tax

affiliation not provided to SSRN

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.

Keywords: Fluid dynamics, Machine learning, Navier-Stokes equations, Physics-informed Neural Network

Suggested Citation

Naderibeni, Mahdi and Reinders, Marcel and Wu, Liang and Tax, David M.J., Learning Solutions of Parametric Navier-Stokes With Physics-Informed Neural Networks. Available at SSRN: https://ssrn.com/abstract=4731727 or http://dx.doi.org/10.2139/ssrn.4731727

Mahdi Naderibeni (Contact Author)

affiliation not provided to SSRN ( email )

No Address Available

Marcel Reinders

Leiden University - Leiden Computational Biology Center ( email )

Netherlands

Liang Wu

affiliation not provided to SSRN ( email )

No Address Available

David M.J. Tax

affiliation not provided to SSRN ( email )

No Address Available

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