Lie Point Symmetry and Physics Informed Networks

TL;DR

This paper introduces a novel method to incorporate Lie point symmetries into physics-informed neural networks (PINNs), enhancing their ability to learn PDE solutions more efficiently by leveraging underlying symmetries.

Contribution

It proposes a symmetry-informed loss function that enforces Lie point symmetries in PINNs, improving their sample efficiency and solution generalization.

Findings

Symmetry loss boosts PINNs' sample efficiency.

Incorporating Lie symmetries improves PDE solution learning.

The approach preserves solution neighborhoods generated by symmetries.

Abstract

Symmetries have been leveraged to improve the generalization of neural networks through different mechanisms from data augmentation to equivariant architectures. However, despite their potential, their integration into neural solvers for partial differential equations (PDEs) remains largely unexplored. We explore the integration of PDE symmetries, known as Lie point symmetries, in a major family of neural solvers known as physics-informed neural networks (PINNs). We propose a loss function that informs the network about Lie point symmetries in the same way that PINN models try to enforce the underlying PDE through a loss function. Intuitively, our symmetry loss ensures that the infinitesimal generators of the Lie group conserve the PDE solutions. Effectively, this means that once the network learns a solution, it also learns the neighbouring solutions generated by Lie point symmetries. Empirical evaluations indicate that the inductive bias introduced by the Lie point symmetries of the PDEs greatly boosts the sample efficiency of PINNs.