Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

TL;DR

Lyapunov-Net is a novel deep neural network architecture designed to efficiently approximate Lyapunov functions for high-dimensional dynamical systems, ensuring positive definiteness and outperforming existing methods.

Contribution

The paper introduces Lyapunov-Net, a neural network architecture with guaranteed positive definiteness and reduced hyper-parameters, along with theoretical analysis and high-dimensional system applications.

Findings

Successfully approximates Lyapunov functions in systems up to 30 dimensions

Outperforms state-of-the-art methods in efficiency and accuracy

Provides theoretical guarantees on approximation power and complexity

Abstract

We develop a versatile deep neural network architecture, called Lyapunov-Net, to approximate Lyapunov functions of dynamical systems in high dimensions. Lyapunov-Net guarantees positive definiteness, and thus it can be easily trained to satisfy the negative orbital derivative condition, which only renders a single term in the empirical risk function in practice. This significantly reduces the number of hyper-parameters compared to existing methods. We also provide theoretical justifications on the approximation power of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the proposed method on nonlinear dynamical systems involving up to 30-dimensional state spaces, and show that the proposed approach significantly outperforms the state-of-the-art methods.