Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers (2410.08304v1)

Abstract: Despite their spectacular progress, LLMs 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.

• The paper introduces a novel transformer-based approach that generates synthetic data to discover global Lyapunov functions.
• The methodology achieves near 100% accuracy and outperforms state-of-the-art solvers by up to tenfold on various systems.
• The models discover new Lyapunov functions in 12.7% of non-polynomial cases, outperforming human experts and opening new research avenues.

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

This paper addresses the complex challenge of discovering Lyapunov functions which ensure the global stability of dynamical systems—a significant open problem in mathematics with no general solution. Traditionally, algorithmic approaches to this problem have been limited to small polynomial systems. The researchers propose a novel method for generating synthetic training samples, utilizing sequence-to-sequence transformers to learn from these datasets. The models trained using this approach outperform traditional algorithmic solvers and human experts on certain benchmarks, even discovering new Lyapunov functions for non-polynomial systems.

Key Contributions and Results

The main contributions of this research are summarized as follows:

1. Synthetic Data Generation: The authors develop a technique to generate training datasets by sampling solutions (Lyapunov functions) synthetically, and subsequently generating systems that fit these solutions. This dual approach, aligned with backward and forward generation strategies, creates a robust dataset for training the transformer models.
2. Model Performance: The transformers achieve high accuracy—near 100% on held-out test sets and substantial out-of-distribution performance. When enriched with a small sample of forward-generated examples, the models generalize effectively across varied datasets.
3. Comparison with State-of-the-Art: The trained models significantly outperform SOSTOOLS, a leading solver for polynomial Lyapunov functions, demonstrating up to tenfold improvement in finding solutions for randomly generated systems.
4. Discovery of New Solutions: On random non-polynomial systems, where no prior algorithmic methods are known, the models discover new Lyapunov functions in 12.7% of test cases, indicating potential breakthroughs in approaching unsolved mathematical problems through AI.
5. Human Comparison: When compared against human performance, specifically a group of mathematics students, the models demonstrate superior accuracy in solving these complex problems, further illustrating their utility and efficiency.

Implications and Future Directions

This research suggests transformative implications for both AI reasoning capabilities and mathematical problem-solving. The success with Lyapunov functions indicates that sequence-to-sequence transformers can tackle other research-level problems in mathematics, potentially shifting how complex mathematical tasks are approached.

From a theoretical perspective, this work supports the notion that transformers can exhibit forms of reasoning or 'intuition' through learning, a critical consideration for advancing AI research. Practically, the methodology outlined could inform the development of tools for researchers across various scientific disciplines, providing AI-derived conjectures or insights for rigorous mathematical problems.

Future developments may aim to scale these models to larger and more intricate systems, exploring richer datasets across broader classes of mathematical functions. The integration of AI in scientific discovery, as evidenced here, marks a pivotal step toward enhancing research methodologies, possibly leading to new mathematical theories and innovations.

Conclusion

The paper presents a methodologically rigorous and technically sophisticated approach to a classic mathematical challenge, yielding superior results using contemporary AI techniques. By paving the way for AI contributions to unsolved problems in mathematics, it underscores the transformative power of neural network models beyond traditional domains, suggesting substantial potential for future breakthroughs in scientific discovery and reasoning.