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

Xiaopei Jiao, Fansheng Xiong

TL;DR

The paper addresses the computational bottleneck of obtaining fundamental solutions for linear PDEs with PINNs by exploiting Lie group symmetries. It introduces the Linearized Symmetric Condition (LSC), $Q = \eta - \xi K_x - \tau K_t = 0$, as a first-order residual that replaces higher-order derivatives in the physics loss, enabling the Deep Symmetric Fundamental Solver (GsPINN). Key contributions include the invariance-principle framework for invariant generators, the LSC-based residual, and numerical demonstrations showing deterministic 20–30% training-time speedups while maintaining accuracy, including high-dimensional problems. This approach offers a scalable, mesh-free pathway for real-time kernel evaluations in Cauchy problems and broadens symmetry-informed methods in Scientific Machine Learning (SciML).

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.

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

TL;DR

The paper addresses the computational bottleneck of obtaining fundamental solutions for linear PDEs with PINNs by exploiting Lie group symmetries. It introduces the Linearized Symmetric Condition (LSC), , as a first-order residual that replaces higher-order derivatives in the physics loss, enabling the Deep Symmetric Fundamental Solver (GsPINN). Key contributions include the invariance-principle framework for invariant generators, the LSC-based residual, and numerical demonstrations showing deterministic 20–30% training-time speedups while maintaining accuracy, including high-dimensional problems. This approach offers a scalable, mesh-free pathway for real-time kernel evaluations in Cauchy problems and broadens symmetry-informed methods in Scientific Machine Learning (SciML).

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.
Paper Structure (13 sections, 7 theorems, 82 equations, 5 figures, 3 tables)

This paper contains 13 sections, 7 theorems, 82 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic concepts and Preliminaries
  3. Physics-Informed neural networks
  4. Cauchy problem
  5. Method
  6. Invariance principle
  7. Deep symmetric fundamental solver
  8. Numerical results
  9. Conclusion
  10. Acknowledgement
  11. Additional background of group theory
  12. Lie symmetry method
  13. Detailed Proofs

Key Result

Theorem 3.1

Assume the first-order infinitesimal generator has the form The generator keeps the following Cauchy problem invariant if and only if where $\text{supp}(f)$ denotes support set of function $f$.

Figures (5)

  • Figure 1: Loss value of vanilla PINN (left) and GsPINN (right) in Example 1.
  • Figure 2: The comparison between PINN and GsPINN with the exact solution in Example 1. (a) We present the predicted solution by PINN, the exact solution, the $L_2$ error, and the predicted symmetry residual, respectively. (b) correspond to GsPINN, with the same meaning.
  • Figure 3: Comparison between exact solution and PINN solution in Example 2.
  • Figure 4: Comparison between the exact solution and GSPINN solution in Example 2.
  • Figure 5: Loss value of GsPINN (left) and vanilla PINN (right) in Example 2.

Theorems & Definitions (29)

  • Definition 2.1: Fundamental solution
  • Definition 3.1
  • Definition 3.2: Distribution
  • Definition 3.3: Transformation of Distributions
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.1
  • Example 4.1: 3D FS of heat equation
  • ...and 19 more