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.
Related
- Linear Lyapunov Functions for Nonlinear Compartmental Systems
- Dynamic Quantum-Assisted Co-Design of Controller and Lyapunov Candidate Parameters for Nonlinear Systems
- Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators
- Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials
- Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification
BibTeX
@misc{mukherjee2026learning,
title = {{Learning Lyapunov Operators for Nonlinear Systems}},
author = {Amartya Mukherjee and Maxwell Fitzsimmons and David C. Del Rey Fernández and Jun Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18894},
eprint = {2609.18894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning Lyapunov Operators for Nonlinear Systems
AU - Amartya Mukherjee
AU - Maxwell Fitzsimmons
AU - David C. Del Rey Fernández
AU - Jun Liu
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18894
UR - https://arxiv.org/abs/2609.18894
AB - 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.
ER -