# Analytical Lyapunov Function Discovery: An RL-based Generative Approach

arXiv: https://arxiv.org/abs/2502.02014 · HTML: https://mlsanity.com/paper/2502.02014/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 31 of its 52 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 30 of 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Deep Generative Symbolic Regression](https://mlsanity.com/paper/2401.00282/) (2023)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Neural Lyapunov Control for Discrete-Time Systems](https://mlsanity.com/paper/2305.06547/) (2023)
- [Controllable Neural Symbolic Regression](https://mlsanity.com/paper/2304.10336/) (2023)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [Reinforcement Learning for Optimal Primary Frequency Control: A Lyapunov Approach](https://mlsanity.com/paper/2009.05654/) (2020)
- [Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints](https://mlsanity.com/paper/2210.00629/) (2022)
- [Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making](https://mlsanity.com/paper/1504.06002/) (2015)
- [Stability Constrained Reinforcement Learning for Decentralized Real-Time Voltage Control](https://mlsanity.com/paper/2209.07669/) (2022)
- [Lyapunov-Regularized Reinforcement Learning for Power System Transient Stability](https://mlsanity.com/paper/2103.03869/) (2021)
- [Bridging Transient and Steady-State Performance in Voltage Control: A Reinforcement Learning Approach with Safe Gradient Flow](https://mlsanity.com/paper/2303.11417/) (2023)
- [Structured Neural-PI Control with End-to-End Stability and Output Tracking Guarantees](https://mlsanity.com/paper/2305.17777/) (2023)
- [Complexity-Aware Deep Symbolic Regression with Robust Risk-Seeking Policy Gradients](https://mlsanity.com/paper/2406.06751/) (2024)
- [Stability-Constrained Learning for Frequency Regulation in Power Grids with Variable Inertia](https://mlsanity.com/paper/2405.20489/) (2024)
- [Online Event-Triggered Switching for Frequency Control in Power Grids with Variable Inertia](https://mlsanity.com/paper/2408.15436/) (2024)
- [Exponential convergence under distributed averaging integral frequency control](https://mlsanity.com/paper/1608.03798/) (2016)
- [A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function](https://mlsanity.com/paper/1803.06087/) (2018)

## Built on by (most cited first, 1 of 1)

- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)

## Abstract (arXiv)

Despite advances in learning-based methods, finding valid Lyapunov functions for nonlinear dynamical systems remains challenging. Current neural network approaches face two main issues: challenges in scalable verification and limited interpretability. To address these, we propose an end-to-end framework using transformers to construct analytical Lyapunov functions (local), which simplifies formal verification, enhances interpretability, and provides valuable insights for control engineers. Our framework consists of a transformer-based trainer that generates candidate Lyapunov functions and a falsifier that verifies candidate expressions and refines the model via risk-seeking policy gradient. Unlike Alfarano et al. (2024), which utilizes pre-training and seeks global Lyapunov functions for low-dimensional systems, our model is trained from scratch via reinforcement learning (RL) and succeeds in finding local Lyapunov functions for high-dimensional and non-polynomial systems. Given the analytical nature of the candidates, we employ efficient optimization methods for falsification during training and formal verification tools for the final verification. We demonstrate the efficiency of our approach on a range of nonlinear dynamical systems with up to ten dimensions and show that it can discover Lyapunov functions not previously identified in the control literature. Full implementation is available on hrefhttps://github.com/JieFeng-cse/Analytical-Lyapunov-Function-DiscoveryGithub