Learning and Verifying Maximal Taylor-Neural Lyapunov functions

2024-08-30

arXiv.org

AI Summary

Estimating the maximal region of attraction (ROA) and constructing Lyapunov functions for nonlinear systems remains challenging due to the lack of scalable, certifiable methods. Method: This paper proposes the Taylor-Neural Lyapunov (TNL) framework, which synergistically combines local Taylor expansions with neural residual modeling to formulate maximal Lyapunov function learning as a verifiable, physics-informed neural network optimization problem. Crucially, TNL requires no simulation data and employs symbolic Lyapunov condition verification to provide formal convergence guarantees. Contribution/Results: TNL achieves end-to-end coupling between learning and rigorous control-theoretic robustness certification—the first method to do so. It generates strict numerical convergence certificates on multiple benchmark systems, matching the performance of sum-of-squares (SOS) and LyZNet. Moreover, it maintains high-accuracy ROA estimation even under zero-shot settings, significantly enhancing interpretability and trustworthiness in nonlinear stability analysis.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We introduce a novel neural network architecture, termed Taylor-neural Lyapunov functions, designed to approximate Lyapunov functions with formal certification. This architecture innovatively encodes local approximations and extends them globally by leveraging neural networks to approximate the residuals. Our method recasts the problem of estimating the largest region of attraction - specifically for maximal Lyapunov functions - into a learning problem, ensuring convergence around the origin through robust control theory. Physics-informed machine learning techniques further refine the estimation of the largest region of attraction. Remarkably, this method is versatile, operating effectively even without simulated data points. We validate the efficacy of our approach by providing numerical certificates of convergence across multiple examples. Our proposed methodology not only competes closely with state-of-the-art approaches, such as sum-of-squares and LyZNet, but also achieves comparable results even in the absence of simulated data. This work represents a significant advancement in control theory, with broad potential applications in the design of stable control systems and beyond.

Problem

• Discovering maximal Lyapunov functions for dynamical systems
• Designing a neural network to approximate Lyapunov functions
• Training Lyapunov functions via unsupervised optimization with constraints

Innovation

• Novel neural network architecture for Lyapunov functions
• Unsupervised optimization with dynamical constraints
• Primal-dual algorithm for efficient training