Problem
Research questions and friction points this paper is trying to address.
Lyapunov functions
nonlinear dynamical systems
stability analysis
partial differential equations
Learning Lyapunov Operators for Nonlinear Systems
🤖 AI Summary
研究通过构建Lyapunov解算子并使用Fourier神经算子来近似解决非线性系统稳定性分析中Lyapunov函数难以构建的问题。
📝 Abstract
Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Fourier Neural Operators
Lyapunov Solution Operator
Nonlinear Systems
Stability Analysis
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