# Combining Neural Networks and Symbolic Regression for Analytical Lyapunov Function Discovery

arXiv: https://arxiv.org/abs/2406.15675 · HTML: https://mlsanity.com/paper/2406.15675/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 32 references that are in the map.
- First posted on arXiv 21 Jun 2024.

## Builds on (most cited first, 20 of 20)

- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [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)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [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)
- [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)
- [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)
- [Compositional Neural Certificates for Networked Dynamical Systems](https://mlsanity.com/paper/2303.14564/) (2023)
- [Stability-Constrained Learning for Frequency Regulation in Power Grids with Variable Inertia](https://mlsanity.com/paper/2405.20489/) (2024)

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

- [Analytical Lyapunov Function Discovery: An RL-based Generative Approach](https://mlsanity.com/paper/2502.02014/) (2025)

## Abstract (arXiv)

We propose CoNSAL (Combining Neural networks and Symbolic regression for Analytical Lyapunov function) to construct analytical Lyapunov functions for nonlinear dynamic systems. This framework contains a neural Lyapunov function and a symbolic regression component, where symbolic regression is applied to distill the neural network to precise analytical forms. Our approach utilizes symbolic regression not only as a tool for translation but also as a means to uncover counterexamples. This procedure terminates when no counterexamples are found in the analytical formulation. Compared with previous results, CoNSAL directly produces an analytical form of the Lyapunov function with improved interpretability in both the learning process and the final results. We apply CoNSAL to 2-D inverted pendulum, path following, Van Der Pol Oscillator, 3-D trig dynamics, 4-D rotating wheel pendulum, 6-D 3-bus power system, and demonstrate that our algorithm successfully finds their valid Lyapunov functions. Code examples are available at https://github.com/HaohanZou/CoNSAL.