(Un)supervised Learning of Maximal Lyapunov Functions

TL;DR

This paper introduces a neural network-based method for discovering maximal Lyapunov functions to determine the region of attraction in dynamical systems, combining local quadratic and global higher-order approximations.

Contribution

It proposes a novel neural network architecture and an unsupervised training algorithm for learning Lyapunov functions, integrating physics-informed learning and data when available.

Findings

Outperforms existing methods in accuracy of attraction regions

Effective for various classes of dynamical systems

Combines local quadratic and global higher-order approximations

Abstract

In this paper, we address the problem of discovering maximal Lyapunov functions, as a means of determining the region of attraction of a dynamical system. To this end, we design a novel neural network architecture, which we prove to be a universal approximator of (maximal) Lyapunov functions. The architecture combines a local quadratic approximation with the output of a neural network, which models global higher-order terms in the Taylor expansion. We formulate the problem of training the Lyapunov function as an unsupervised optimization problem with dynamical constraints, which can be solved leveraging techniques from physics-informed learning. We propose and analyze a tailored training algorithm, based on the primal-dual algorithm, that can efficiently solve the problem. Additionally, we show how the learning problem formulation can be adapted to integrate data, when available. We apply the proposed approach to different classes of systems, showing that it matches or outperforms state-of-the-art alternatives in the accuracy of the approximated regions of attraction.