Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known gen...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Alfarano, Alberto, Charton, François, Hayat, Amaury
Format: Article
Language:English
Subjects:
Dynamical systems
Liapunov functions
Polynomials
Solvers
Task complexity
Transformers
Online Access:Get full text
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Summary:Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.
ISSN:2331-8422