# Learning Lyapunov Operators for Nonlinear Systems

arXiv: https://arxiv.org/abs/2609.18894 · HTML: https://mlsanity.com/paper/2609.18894/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 13 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization](https://mlsanity.com/paper/1703.06868/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Physics-Informed Neural Network Policy Iteration: Algorithms, Convergence, and Verification](https://mlsanity.com/paper/2402.10119/) (2024)
- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)

## Abstract (arXiv)

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.