Leveraging Lie Group Symmetries to Enhance Physics-Informed Neural Networks for the Fundamental Solution of Linear PDEs

TL;DR

This paper introduces a novel method combining Lie group symmetries with physics-informed neural networks to efficiently solve linear PDEs, reducing computational costs and enhancing accuracy without needing prior solutions.

Contribution

It presents a new framework integrating Lie symmetries into PINNs, including an invariant generator identification algorithm that improves efficiency and solution quality.

Findings

Significant reduction in training time for PDE solutions

Maintained high accuracy with fewer derivatives in the loss function

Framework applicable to arbitrary Cauchy problems

Abstract

Since the introduction of deep learning for solving partial differential equations (PDEs), there has been growing interest in real-time system responses, where the kernel function plays a key role. Physics-informed neural networks (PINNs), a popular mesh-free, semi-supervised learning tool, offer high flexibility. This paper explores the integration of Lie symmetry groups with deep learning techniques to enhance the numerical solutions of fundamental PDEs. We propose a novel approach that combines PINNs and Lie group theory to address computational inefficiencies in traditional methods. By incorporating the linearized symmetric condition (LSC) derived from Lie symmetries into PINNs, we introduce a new residual loss function that requires fewer derivatives for calculation. This integration reduces computational costs and improves solution accuracy. Numerical simulations demonstrate a significant reduction in training time while maintaining accuracy. Additionally, we provide a framework for identifying invariant infinitesimal generators for arbitrary Cauchy problems. This unsupervised algorithm does not require prior numerical solutions, making it practical and efficient for various applications.