Research on Symmetry Enhancement of Physics-Informed Neural Networks Based on Lie Group Theory

This study explores four strategies for enforcing symmetry in Physics-Informed Neural Networks (PINNs) using Lie group theory, verifies them experimentally on steady-state heat conduction and Helmholtz equations, and finds that architecture encoding and activation function design are complementary key elements.

Physics-Informed Neural Networks (PINNs) embed physical laws into loss functions and are suitable for solving partial differential equations (PDEs). However, standard PINNs lack an inherent mechanism to leverage the symmetry of physical problems. Symmetry is a fundamental concept in physics, corresponding to conservation laws, which can reduce computational cost, improve solution accuracy and stability (e.g., rotationally symmetric problems can be solved with dimensionality reduction).

Using the theoretical framework of Lie groups (e.g., SO(2) rotation group) and Lie algebras, symmetry is introduced through Lie derivative constraints or architectural design. The four strategies are: 1. Full-region baseline (no constraints, control group); 2. Quarter-region + Neumann boundary conditions (dimensionality reduction); 3. Quarter-region + soft penalty of Lie derivatives; 4. Full-region architecture encoding (output is the average of four rotated copies, naturally satisfying symmetry).

Steady-state heat conduction: The architecture encoding strategy achieves the best accuracy (relative L2 error of 0.0018%); the quarter-region method reduces training time by about 44%; all enhancement strategies outperform the baseline. Helmholtz equation: Training with tanh activation function does not converge (vanishing gradients); replacing it with a sine function (SIREN) reduces the error to 0.85%, but the PDE residual still stagnates, requiring improved optimization strategies.

Symmetry enhancement and activation function design are complementary needs: Symmetry constraints alone are insufficient (e.g., tanh's vanishing gradients), and a good activation function without symmetry guidance wastes resources. Only their combination can effectively improve PINN performance.

For the optimization pathologies of the Helmholtz equation, we can explore: adaptive loss weight scheduling, curriculum learning to gradually increase PDE constraints, spectral methods for oscillatory solutions, and hybrid numerical-neural network methods.

The project provides complete code that can run on Google Colab: Four strategies for the heat conduction equation correspond to independent Notebooks; the Helmholtz equation is integrated into an interactive Notebook for easy configuration switching, ensuring reproducibility and extensibility.

This study combines Lie group theory with machine learning to solve scientific computing problems. It can be extended to engineering scenarios with symmetry (e.g., turbine rotational symmetry, crystal discrete symmetry). The PINN field still needs to explore directions such as activation functions, optimization algorithms, and multi-scale processing.