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.
Amartya Mukherjee, Maxwell Fitzsimmons, D. C. Del Rey Fernández et al.· 0 citations
Natural-language descriptions of optimization problems may be incomplete or vague about numerical information that a solver requires, including costs, capacities, demands, bounds, and penalties. A language model can translate the description into code, but when a required value is absent it must either stop or guess. We present SAILOR, a proof-of-concept system that detects such unsupported numerical choices, asks the user targeted follow-up questions, and updates the optimization model before returning a solution. Questions are prioritized using uncertainty and solver-derived estimates of how strongly each missing value affects the current model. We evaluate the pipeline on 1,723 instances from seven masked benchmarks using an idealized simulator that returns ground-truth values. Exact objective-value agreement ranges from 27.0% to 87.6% across datasets, with 1.4--5.7 questions per instance on average. These results establish feasibility under controlled branch-and-reveal feedback; they do not measure performance with human users or general structural model repair. Code is available at: https://github.com/sshaghayeghs/SAILOR.
Shaghayegh Sadeghi, Steve Smith, D. C. Del Rey Fernández· 0 citations
It is proved that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution.
Amartya Mukherjee, Maxwell Fitzsimmons, D. C. Del Rey Fernández et al.· arXiv.org· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
