Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks

TL;DR

This paper introduces a projection method for Physics-Informed Neural Networks that ensures the exact conservation of integral quantities, significantly improving physical law adherence without sacrificing PDE solution accuracy.

Contribution

A novel projection technique for PINNs that guarantees conservation of integral quantities, enhancing physical fidelity and convergence.

Findings

Reduced conservation error by 3-4 orders of magnitude

Marginally improved PDE solution accuracy

Enhanced convergence through better loss landscape conditioning

Abstract

We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.