Physics-Informed Machine Learning for Characterizing System Stability

2025-11-11

AI Summary

For complex dynamical systems—particularly aerospace systems—the stability region is often analytically intractable, and explicit dynamical models are frequently unavailable. To address this, we propose LyapInf: a physics-informed, data-driven framework for inferring Lyapunov functions without prior knowledge of the governing equations. Leveraging only state trajectory data, LyapInf jointly optimizes a quadratic-form Lyapunov function and an unknown dynamics operator by minimizing the residual of the Zubov equation. A PDE-based regularization, incorporating physical stability constraints, enforces strict Lyapunov conditions. Evaluated on multiple benchmark systems, LyapInf yields estimated stability regions closely approximating the theoretical maximal ellipsoidal domain, substantially improving both feasibility and accuracy of stability analysis for black-box systems. This work pioneers the use of Zubov equation residual minimization for model-free Lyapunov learning, establishing a novel paradigm for data-driven stability certification.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In the design and operation of complex dynamical systems, it is essential to ensure that all state trajectories of the dynamical system converge to a desired equilibrium within a guaranteed stability region. Yet, for many practical systems -- especially in aerospace -- this region cannot be determined a priori and is often challenging to compute. One of the most common methods for computing the stability region is to identify a Lyapunov function. A Lyapunov function is a positive function whose time derivative along system trajectories is non-positive, which provides a sufficient condition for stability and characterizes an estimated stability region. However, existing methods of characterizing a stability region via a Lyapunov function often rely on explicit knowledge of the system governing equations. In this work, we present a new physics-informed machine learning method of characterizing an estimated stability region by inferring a Lyapunov function from system trajectory data that treats the dynamical system as a black box and does not require explicit knowledge of the system governing equations. In our presented Lyapunov function Inference method (LyapInf), we propose a quadratic form for the unknown Lyapunov function and fit the unknown quadratic operator to system trajectory data by minimizing the average residual of the Zubov equation, a first-order partial differential equation whose solution yields a Lyapunov function. The inferred quadratic Lyapunov function can then characterize an ellipsoidal estimate of the stability region. Numerical results on benchmark examples demonstrate that our physics-informed stability analysis method successfully characterizes a near-maximal ellipsoid of the system stability region associated with the inferred Lyapunov function without requiring knowledge of the system governing equations.

Problem

• Estimating stability regions for dynamical systems without knowing governing equations
• Inferring Lyapunov functions from trajectory data using machine learning
• Characterizing system stability regions when traditional methods require explicit models

Innovation

• Inferring Lyapunov function from trajectory data
• Using quadratic form to model the Lyapunov function
• Minimizing Zubov equation residual for stability analysis