Lie Group-Based Neural Networks for Nonlinear Dynamics

Abstract

This paper introduces a novel neural network approach based on Lie groups to effectively solve initial value problems of differential equations for nonlinear dynamical systems. Our method utilizes a priori knowledge inherent in the system, i.e. Lie group expressions, and employs a single-layer network structure with the essence of a multilayer perceptron (MLP). To validate the effectiveness of our approach, we conducted an extensive empirical study using various examples representing complex nonlinear dynamical systems. The research results demonstrate the outstanding performance and efficacy of our method, outperforming Neural Ordinary Differential Equations in terms of accuracy, convergence speed, and stability.