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Hard Constraint Projection in a Physics Informed Neural Network

Miranda J. S. Horne, Peter K. Jimack, Amirul Khan, He Wang

TL;DR

The paper addresses enforcing exact discretised physics in neural surrogates for nonlinear PDEs by extending hard constraint projection (HCP) to the 2D incompressible Navier–Stokes equations within a PINN framework. It uses a stream function $\psi$ and pressure $p$ to encode the velocity field with $u=\partial\psi/\partial y$ and $v=-\partial\psi/\partial x$, while projecting predictions onto the discretised NS hyperplane via $P=I-A^T(AA^T)^{-1}A$ to satisfy $AB=0$. Empirical results show similar training behavior between PINN and HCP-PINN, with no consistent reduction in the physics error for the HCP and higher computational costs, highlighting potential limitations for nonlinear PDEs and the need for discretisation improvements. The work outlines future directions including boundary-condition treatment via ghost cells, robustness to noisy data, and extending the method to broader flow regimes, emphasizing careful consideration of when HCP offers practical gains for nonlinear systems.

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.

Hard Constraint Projection in a Physics Informed Neural Network

TL;DR

The paper addresses enforcing exact discretised physics in neural surrogates for nonlinear PDEs by extending hard constraint projection (HCP) to the 2D incompressible Navier–Stokes equations within a PINN framework. It uses a stream function and pressure to encode the velocity field with and , while projecting predictions onto the discretised NS hyperplane via to satisfy . Empirical results show similar training behavior between PINN and HCP-PINN, with no consistent reduction in the physics error for the HCP and higher computational costs, highlighting potential limitations for nonlinear PDEs and the need for discretisation improvements. The work outlines future directions including boundary-condition treatment via ghost cells, robustness to noisy data, and extending the method to broader flow regimes, emphasizing careful consideration of when HCP offers practical gains for nonlinear systems.

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.
Paper Structure (5 sections, 6 equations, 3 figures)

This paper contains 5 sections, 6 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PHYSICS INFORMED NEURAL NETWORK (PINN)
  3. HARD CONSTRAINT PROJECTION (HCP)
  4. PROVISIONAL RESULTS
  5. CURRENT WORK

Figures (3)

  • Figure 1: HCP-PINN architecture. The left box is the FFNN with learnable weights and biases, and the right box is the unlearnable HCP.
  • Figure 2: Graphs comparing the value of the loss, data error, and physics error, on the test and training set, during the training of both models.
  • Figure 3: Ground truth velocity field, and corresponding model predictions at epoch 750, of the velocity field at $t=4.8$.