Hard Constraint Projection in a Physics Informed Neural Network

TL;DR

This paper introduces a method to embed hard constraints into physics informed neural networks for solving complex non-linear PDEs like the Navier-Stokes equations, improving solution accuracy and physical consistency.

Contribution

It extends the hard constraint projection method to non-linear PDEs within PINNs, enabling exact solutions for fluid dynamics problems.

Findings

Successfully applied to 2D incompressible Navier-Stokes equations

Enforces exact PDE solutions via a novel projection layer

Improves physical fidelity of neural network predictions

Abstract

In this work, we embed hard constraints in a physics informed neural network (PINN) which predicts solutions to the 2D incompressible Navier Stokes equations. We extend the hard constraint method introduced by Chen et al. (arXiv:2012.06148) from a linear PDE to a strongly non-linear PDE. The PINN is used to estimate the stream function and pressure of the fluid, and by differentiating the stream function we can recover an incompressible velocity field. An unlearnable hard constraint projection (HCP) layer projects the predicted velocity and pressure to a hyperplane that admits only exact solutions to a discretised form of the governing equations.