{"ID":22952758,"CreatedAt":"2026-09-17T02:12:05.498442134Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18894","arxiv_id":"2609.18894","title":"Learning Lyapunov Operators for Nonlinear Systems","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.","short_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 th...","url_abs":"https://arxiv.org/abs/2609.18894","url_pdf":"https://arxiv.org/pdf/2609.18894v1","authors":"[\"Amartya Mukherjee\",\"Maxwell Fitzsimmons\",\"David C. Del Rey Fernández\",\"Jun Liu\"]","published":"2026-09-16T16:28:25Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}