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

arXiv: https://arxiv.org/abs/2410.08304 · HTML: https://mlsanity.com/paper/2410.08304/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 17 of its 44 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 17 of 17)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Large Language Models Are Human-Level Prompt Engineers](https://mlsanity.com/paper/2211.01910/) (2022)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Investigating the Limitations of Transformers with Simple Arithmetic Tasks](https://mlsanity.com/paper/2102.13019/) (2021)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [Robustness of Control Barrier Functions for Safety Critical Control](https://mlsanity.com/paper/1612.01554/) (2016)
- [It's Not What Machines Can Learn, It's What We Cannot Teach](https://mlsanity.com/paper/2002.09398/) (2020)
- [SALSA: Attacking Lattice Cryptography with Transformers](https://mlsanity.com/paper/2207.04785/) (2022)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [Formalising perfectoid spaces](https://mlsanity.com/paper/1910.12320/) (2019)
- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (2024)
- [Classical Converse Theorems in Lyapunov's Second Method](https://mlsanity.com/paper/1502.04809/) (2015)

## Built on by (most cited first, 29 of 29)

- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [AIGS: Generating Science from AI-Powered Automated Falsification](https://mlsanity.com/paper/2411.11910/) (2024)
- [Auto-Formulating Dynamic Programming Problems with Large Language Models](https://mlsanity.com/paper/2507.11737/) (2025)
- [LemmaBench: A Live, Research-Level Benchmark to Evaluate LLM Capabilities in Mathematics](https://mlsanity.com/paper/2602.24173/) (2026)
- [CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs](https://mlsanity.com/paper/2502.18663/) (2025)
- [Transforming Calabi-Yau Constructions: Generating New Calabi-Yau Manifolds with Transformers](https://mlsanity.com/paper/2507.03732/) (2025)
- [Learning Interpretable Network Dynamics via Universal Neural Symbolic Regression](https://mlsanity.com/paper/2411.06833/) (2024)
- [Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?](https://mlsanity.com/paper/2501.18527/) (2025)
- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)
- [Geometric Generality of Transformer-Based Gr"obner Basis Computation](https://mlsanity.com/paper/2504.12465/) (2025)
- [From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm](https://mlsanity.com/paper/2511.12421/) (2025)
- [Flow-based Extremal Mathematical Structure Discovery](https://mlsanity.com/paper/2601.18005/) (2026)
- [Can Neural Networks Learn Small Algebraic Worlds? An Investigation Into the Group-theoretic Structures Learned By Narrow Models Trained To Predict Group Operations](https://mlsanity.com/paper/2601.21150/) (2026)
- [Local Look-Ahead Guidance via Verifier-in-the-Loop for Automated Theorem Proving](https://mlsanity.com/paper/2503.09730/) (2025)
- [Analytical Lyapunov Function Discovery: An RL-based Generative Approach](https://mlsanity.com/paper/2502.02014/) (2025)
- [Int2Int: a framework for mathematics with transformers](https://mlsanity.com/paper/2502.17513/) (2025)
- [Counting Cycles with AI: Counting Cycles with AI: Computationally Efficient Equivalent Forms with Applications](https://mlsanity.com/paper/2505.17964/) (2025)
- [Discovering Learning-Friendly Generation Orders for Sequential Computation](https://mlsanity.com/paper/2506.23875/) (2025)
- [CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)](https://mlsanity.com/paper/2509.19162/) (2025)
- [Mapping Uncharted Symmetries: Machine Discovery in Combinatorics](https://mlsanity.com/paper/2605.19063/) (2026)
- [CALT: A Library for Computer Algebra with Transformer](https://mlsanity.com/paper/2506.08600/) (2025)
- [Learning Moderately Input-Sensitive Functions: A Case Study in QR Code Decoding](https://mlsanity.com/paper/2506.20305/) (2025)
- [Breaking the Data Barrier in Learning Symbolic Computation: A Case Study on Variable Ordering Suggestion for Cylindrical Algebraic Decomposition](https://mlsanity.com/paper/2601.13731/) (2026)
- [Approximating Matrix Functions with Deep Neural Networks and Transformers](https://mlsanity.com/paper/2602.07800/) (2026)
- [Reconstructing conformal field theoretical compositions with Transformers](https://mlsanity.com/paper/2605.01072/) (2026)
- [Learning Large-Scale Modular Addition with an Auxiliary Modulus](https://mlsanity.com/paper/2605.07648/) (2026)
- [Learning First Integrals via Backward-Generated Data and Guided Reinforcement Learning](https://mlsanity.com/paper/2605.21160/) (2026)
- [Generating Special Triangulations with Transformers](https://mlsanity.com/paper/2606.26660/) (2026)

## Abstract (arXiv)

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.