(Un)supervised Learning of Maximal Lyapunov Functions

TL;DR

The paper tackles the problem of identifying the region of attraction for nonlinear dynamical systems by learning a maximal Lyapunov function $V^*$ that certifies stability. It introduces Taylor-Neural Lyapunov functions, a neural architecture that blends a local quadratic form with neural residuals modeling higher-order Taylor terms, and trains them through physics-informed unsupervised learning, with a primal-dual algorithm and boundary-estimation techniques. A universal approximation theorem is established for the architecture, and a supervised variant incorporating simulator data is also developed. Across multiple dynamical scenarios including high-dimensional and stiff systems, the method yields ROA estimates that match or surpass state-of-the-art approaches, demonstrating robustness to initialization and scalability to complex dynamics.

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.

Figures (4)

• Figure 1: Phase-plane interpretation of Zubov's theorem. The point $x_0$ is on the boundary of a suboptimal basin of attraction with Lyapunov function $V$. The flow (red arrow) is entering the level set. Along the region of attraction, at point $x_1$, the flow (red arrow) is perpendicular to the gradient (green arrow).
• Figure 2: Unsupervised Taylor-neural Lyapunov function for globally stable system \ref{['eq:globally_stable2']}. The color scale represents the sub-level sets of the Lyapunov function. The estimated region of attraction is colored. Blue dots refer to the sampling points at the boundary $\{\eta_i x_i\}_i$. Arrows indicate the flow of the original system.
• Figure 3: Supervised Taylor-neural Lyapunov function for locally stable system \ref{['eq:locally_stable']}. The color scale represents the sub-level sets of the Lyapunov function. The estimated region of attraction is the colored area. Blue dots refer to the sampling points at the boundary $\{\eta_i x_i\}_i$. Arrows indicate the flow of the original system.
• Figure 4: Taylor-neural Lyapunov function for Van der Pol oscillator with $\mu = 1$. The color scale represents the sub-level sets of the Lyapunov function. The estimated region of attraction is the colored area. Blue dots refer to the sampling points at the boundary $\{\eta_i x_i\}_i$. Arrows indicate the flow of the original system. Dash-line is the estimated region of attraction obtained using SOS.

Theorems & Definitions (34)

• Definition 1: Local asymptotic stability glad2018control
• Definition 2: Basin and region of attraction khalil2002nonlinear
• Definition 3: Lyapunov function khalil2002nonlinear
• Theorem 1: Lyapunov characterization of basin of attr.
• Theorem 2: Lyapunov characterization of region of attr.
• Lemma 1: $V^*$ around the origin
• proof
• Definition 4: Neural network residual
• Remark 1
• Definition 5: Taylor-neural Lyapunov function