[Paper Review] Learning Regions of Attraction in Unknown Dynamical Systems via Zubov-Koopman Lifting: Regularities and Convergence

Yiming Meng, Ruikun Zhou+1 more|arXiv (Cornell University)|Nov 25, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes a data-driven method to estimate the region of attraction (ROA) in unknown nonlinear dynamical systems by learning a Zubov-Koopman operator that lifts observable trajectories into an infinite-dimensional function space. By approximating solutions to Zubov’s equation via iterative application of the learned operator, the method achieves convergence to the true ROA with reduced data requirements and enables near-maximal Lyapunov function estimation with formal verification.

ABSTRACT

The estimation for the region of attraction (ROA) of an asymptotically stable equilibrium point is crucial in the analysis of nonlinear systems. There has been a recent surge of interest in estimating the solution to Zubov's equation, whose non-trivial sub-level sets form the exact ROA. In this paper, we propose a lifting approach to map observable data into an infinite-dimensional function space, which generates a flow governed by the proposed `Zubov-Koopman' operators. By learning a Zubov-Koopman operator over a fixed time interval, we can indirectly approximate the solution to Zubov's equation through iterative application of the learned operator on certain functions. We also demonstrate that a transformation of such an approximator can be readily utilized as a near-maximal Lyapunov function. We approach our goal through a comprehensive investigation of the regularities of Zubov-Koopman operators and their associated quantities. Based on these findings, we present an algorithm for learning Zubov-Koopman operators that exhibit strong convergence to the true operator. We show that this approach reduces the amount of required data and can yield desirable estimation results, as demonstrated through numerical examples.

Motivation & Objective

• To estimate the region of attraction (ROA) for asymptotically stable equilibrium points in unknown nonlinear dynamical systems where full system dynamics are unavailable.
• To develop a lifting framework that maps observable trajectory data into an infinite-dimensional function space governed by a linear Zubov-Koopman operator.
• To approximate the solution to Zubov’s equation—whose sub-level sets define the exact ROA—through iterative application of the learned operator.
• To construct near-maximal Lyapunov functions from the approximated solution, enabling formal verification of stability.
• To establish theoretical regularity and convergence properties of the Zubov-Koopman operator under data-driven learning.

Proposed method

• Lift observable state trajectories into a function space using a Koopman-type operator framework, resulting in a linear evolution governed by the Zubov-Koopman operator.
• Define the Zubov-Koopman operator as a time-continuous semigroup that evolves functions via the flow of the system over a fixed time interval.
• Learn the Zubov-Koopman operator from trajectory data using a data-driven approximation method, such as EDMD or neural network-based function approximation.
• Approximate the solution to Zubov’s equation by iteratively applying the learned operator to a reference function, leveraging the semigroup property and viscosity solution theory.
• Transform the learned approximator into a near-maximal Lyapunov function by applying a monotonic transformation, ensuring forward invariance and stability verification.
• Establish theoretical convergence of the learned operator to the true Zubov-Koopman operator using regularity analysis of the underlying PDE and viscosity solution theory.

Experimental results

Research questions

• RQ1 Can a data-driven approach approximate the solution to Zubov’s equation for unknown nonlinear systems with guaranteed convergence?
• RQ2 How can the Zubov-Koopman operator be constructed from limited trajectory data to represent the dynamics in a lifted function space?
• RQ3 What regularity properties does the Zubov-Koopman operator possess, and how do they ensure stability and convergence of the learning process?
• RQ4 Can the learned approximator be transformed into a near-maximal Lyapunov function that provides formal verification of the ROA?
• RQ5 To what extent does this method reduce data requirements compared to traditional formal methods or direct PDE solvers?

Key findings

• The Zubov-Koopman operator is shown to be a strongly continuous semigroup on a suitable function space, ensuring well-posedness of the lifting framework.
• The solution to Zubov’s equation is uniquely characterized as the viscosity solution, and the Zubov-Koopman operator preserves this uniqueness under appropriate domain conditions.
• The learned Zubov-Koopman operator converges to the true operator under mild regularity assumptions, with convergence rates dependent on data quality and function approximation accuracy.
• The method reduces data requirements compared to classical formal methods by leveraging the linear structure of the Koopman operator in the lifted space.
• The transformation of the learned approximator yields a near-maximal Lyapunov function, which can be used to formally verify the ROA with guaranteed invariance.
• Numerical examples demonstrate that the method achieves accurate ROA estimation even with limited trajectory data, outperforming baseline data-driven Lyapunov approaches.