Learning Maximal Regions of Attraction on Compact Sets using Taylor Lyapunov Networks — arXiv eess.SY

Learning Maximal Regions of Attraction on Compact Sets using Taylor Lyapunov Networks — arXiv eess.SY

Matthieu Barreau and Nicola Bastianello propose a neural-network method to estimate the maximal region of attraction: the states from which a nonlinear dynamical system approaches its equilibrium. They identify two obstacles in existing approaches: finite-width networks may fail to preserve strict Lyapunov inequalities near the equilibrium, and bounded-domain training does not directly fit the classical Lyapunov and Zubov theorems. The authors derive versions of those theorems for a prescribed compact domain, where a boundary-value PDE characterizes the maximal region. Their Taylor Lyapunov network sets a quadratic leading-order term using the system’s linearization and represents the higher-order remainder through a structured cubic factorization. They say this construction makes each network in the class a valid local Lyapunov function by construction, and prove the class can approximate maximal Lyapunov functions on compact domains. They combine these results in an unsupervised, physics-informed training procedure. The abstract reports numerical experiments across systems with varying dimensions and stability types, and says the architecture is efficient at estimating the maximum region of attraction. It gives no experiment-specific results, system details or measure of efficiency, so the reported efficiency’s scale and conditions cannot be assessed from this evidence.

Publication: 5f8ed4ba-bda1-4ad2-8679-21c915c9df4d

attribution — pass: The reading correctly credits Matthieu Barreau and Nicola Bastianello. Abstract statements are framed as the authors' claims or as what the abstract reports.

Authors:Matthieu Barreau, Nicola Bastianello

summary — uncertain: The reading is faithful and explains region of attraction. However, several central terms are never explained in plain language: Lyapunov function, strict Lyapunov inequalities, Zubov theorems, boundary-value PDE, compact domain, linearization and physics-informed training. Without an explanation of what a Lyapunov function does, which is to certify stability, a non-specialist cannot follow the core idea or why a guarantee by construction matters. Smallest repair: add a one-line plain explanation of a Lyapunov function, or term-library tooltips for it and the Zubov theorems.

We consider the problem of learning maximal Lyapunov functions to compute the maximal region of attraction of nonlinear dynamical systems using neural networks.

date — pass: originalPublished is 2024-08-30, which matches the v1 submission date. The later v3 revision and the feed date are correctly not used.

Submitted on 30 Aug 2024 (v1), last revised 29 Oct 2026 (this version, v3)

support — pass: Each claim matches the abstract: the two obstacles, the compact-domain theorems with the PDE characterization, the architecture, the local-validity guarantee, universal approximation and the experiments. The caveat that no efficiency measure is given is accurate. The question follows from the evidence without assuming an answer.

we prove that Taylor Lyapunov networks are universal approximators of maximal Lyapunov functions on compact domains

media — pass: The selected media is the verified arXiv abstract page for this paper, which is the direct original source.