Physics-Informed Machine Learning for Characterizing System Stability
Tomoki Koike, Elizabeth Qian
TL;DR
The paper tackles the challenge of estimating the stability region for nonlinear dynamical systems without explicit governing equations by proposing LyapInf, a physics-informed method that infers a quadratic Lyapunov function from trajectory data by minimizing the Zubov residual. By restricting to a quadratic form and embedding the Zubov PDE into a convex optimization, LyapInf yields an interpretable ellipsoidal estimate of the stability region that does not require knowledge of $f$. Numerical experiments across six benchmark systems demonstrate near-maximal ellipsoidal regions inside the true stability domain and show favorable data efficiency compared to neural-network-based approaches. The approach offers a practical, interpretable tool for safety-certified operation in black-box or partially unknown systems, with potential extensions to higher-order Lyapunov functions and high-dimensional models.
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.