研究用神经网络近似非线性系统李雅普诺夫函数
Learning Lyapunov Operators for Nonlinear Systems
这是关于如何用神经网络来近似非线性系统李雅普诺夫函数的研究，对做控制理论或系统稳定性分析的人可能有参考价值。
这篇论文提出了一种方法，通过傅里叶神经算子（FNOs）来近似李雅普诺夫算子，该算子可以将一个向量场映射到对应的李雅普诺夫函数。研究证明了在吸引域的紧子集上，这个算子是良好定义的、唯一的且连续的。数值实验表明，一个训练好的算子可以准确地近似参数化动力学族中的数值李雅普诺夫函数。
Learning Lyapunov Operators for Nonlinear Systems
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.