Neural operators approximate stability functions for nonlinear systems
Learning Lyapunov Operators for Nonlinear Systems
Machine Learning
Summary
Stability analysis for nonlinear systems is tricky because it requires special functions called Lyapunov functions, which are hard to find and usually unique to each system. The authors studied a general way to link system behaviors to these functions through a mathematical operator that works reliably across similar systems. They used neural networks designed to learn operators to approximate this link, enabling one trained system to predict Lyapunov functions for many related nonlinear systems. Their experiments show this approach can accurately replicate traditional calculations for these stability functions.
What this means in practice
- •For control system engineers: Estimate stability conditions efficiently for varying nonlinear system designs using a single trained neural operator.
- •For robotics software developers: Incorporate learned Lyapunov function approximations to verify stability across parameter changes in robot dynamics models.
Authors
Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández, Jun Liu
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.