A Neural Network Approach to Finding Global Lyapunov Functions for Homogeneous Vector Fields

Homogeneous vector fields play an important role in approximating general nonlinear systems. We propose a neural network approach for finding global Lyapunov functions for homogeneous vector fields. We first establish a universal approximation result that guarantees the existence of neural Lyapunov functions for homogeneous systems. We then propose a specific neural network structure for computing such Lyapunov functions, with verifiable sufficient conditions for global asymptotic stability. We demonstrate the potential advantage of neural Lyapunov functions over sums-of-squares Lyapunov functions through numerical examples.