Computing Lyapunov functions using deep neural networks

TL;DR

This paper introduces a deep neural network approach for approximating Lyapunov functions in nonlinear systems, effectively addressing high-dimensional challenges and demonstrating promising results in systems up to ten dimensions.

Contribution

It presents a novel neural network architecture and training method for Lyapunov functions, overcoming the curse of dimensionality under certain system conditions.

Findings

Polynomial growth in neurons needed for approximation

Applicable to systems with compositional Lyapunov functions

Successful numerical examples up to ten dimensions

Abstract

We propose a deep neural network architecture and a training algorithm for computing approximate Lyapunov functions of systems of nonlinear ordinary differential equations. Under the assumption that the system admits a compositional Lyapunov function, we prove that the number of neurons needed for an approximation of a Lyapunov function with fixed accuracy grows only polynomially in the state dimension, i.e., the proposed approach is able to overcome the curse of dimensionality. We show that nonlinear systems satisfying a small-gain condition admit compositional Lyapunov functions. Numerical examples in up to ten space dimensions illustrate the performance of the training scheme.