Enhancing PINN Performance Through Lie Symmetry Group

TL;DR

This work targets improved PDE solving with Physics-Informed Neural Networks (PINNs) by embedding Lie symmetry group structure through infinitesimal generators. Using Burgers' equation as a test case, the authors develop a progression of models: Case A (baseline PINN), Case B (m-SPINN) incorporating transformed collocation points and a symmetry residual, and Case C (m-ASPINN) adding adaptive activation. The approach yields substantial accuracy gains and computational efficiency, achieving results competitive with established numerical methods such as MCB-DQM and WA-DQM. The study demonstrates that integrating abstract symmetry concepts with adaptive deep learning mechanisms can significantly advance data-driven scientific computing.

Abstract

This paper presents intersection of Physics informed neural networks (PINNs) and Lie symmetry group to enhance the accuracy and efficiency of solving partial differential equation (PDEs). Various methods have been developed to solve these equations. A Lie group is an efficient method that can lead to exact solutions for the PDEs that possessing Lie Symmetry. Leveraging the concept of infinitesimal generators from Lie symmetry group in a novel manner within PINN leads to significant improvements in solution of PDEs. In this study three distinct cases are discussed, each showing progressive improvements achieved through Lie symmetry modifications and adaptive techniques. State-of-the-art numerical methods are adopted for comparing the progressive PINN models. Numerical experiments demonstrate the key role of Lie symmetry in enhancing PINNs performance, emphasizing the importance of integrating abstract mathematical concepts into deep learning for addressing complex scientific problems adequately.

Figures (15)

• Figure 1: Physics Informed Neural Network for Burgers' Equation
• Figure 2: Absolute Error Comparison between m-ASPINN, WA-DQM,and MCB-DQM
• Figure 3: Physical behaviour of m-ASPINN approximated solution of u(x,t) at v = 0.1 (right) and series solution (left).
• Figure 4: Cross-section curves by m-ASPINN approximated solution of u(x,t) at v = 0.1 (right) at t= 0, 0.4, 0.8 and 1 and series solution approximated cross-section curves (left).
• Figure 5: Gradient lines plot by m-ASPINN method (right) and by series solution (left) for solution surfaces as shown in figure \ref{['ajk']}