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

Alberto Alfarano, François Charton, Amaury Hayat

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 32 sections, 4 theorems, 28 equations, 2 figures, 14 tables.

Table of Contents

  1. Introduction
  2. System stability and Lyapunov functions
  3. Experimental settings
  4. Data generation
  5. Backward generation
  6. Forward generation
  7. Datasets
  8. Results
  9. In and out-of-distribution accuracy
  10. Enriching training distributions for improved performance
  11. Comparing with state-of-the-art baselines
  12. Into the wild - discovering new mathematics
  13. Expert iteration
  14. Discussion
  15. Mathematical definitions
  16. ...and 17 more sections

Key Result

Theorem 2.3

If the system eq:sys1 has a Lyapunov function, then it is stable.

Figures (2)

  • Figure 1: Dynamic of a stable system: trajectories may be complicated but as long as they start in the red ball they remain in the blue ball.
  • Figure 2: Two stable systems and associated Lyapunov functions discovered by our model. The second, a polynomial system with a non-polynomial Lyapunov function, was studied in KrsticParrilo.

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Lyapunov 1892
  • Theorem 2.4: LaSalle, 1961
  • Definition A.1
  • Theorem A.2: LaSalle Invariance Principle (global)
  • Definition A.3
  • Definition A.4
  • Theorem A.5: LaSalle Invariance Principle (local)