Generalized Lie Symmetries in Physics-Informed Neural Operators

TL;DR

This paper introduces a novel loss augmentation method for physics-informed neural operators that utilizes generalized Lie symmetries, specifically evolutionary representatives, to improve training efficiency and accuracy in solving PDEs.

Contribution

It proposes a new approach that leverages generalized symmetries to enhance neural operator training, addressing limitations of standard symmetry methods.

Findings

Enhanced training efficiency and accuracy of neural operators.

Improved data efficiency in learning PDE solution operators.

Demonstrated effectiveness on various PDE problems.

Abstract

Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.