# Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification

arXiv: https://arxiv.org/abs/2312.09131 · HTML: https://mlsanity.com/paper/2312.09131/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 61 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 15 of 15)

- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [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)
- [Learning Lyapunov Functions for Hybrid Systems](https://mlsanity.com/paper/2012.12015/) (2020)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [Learning Region of Attraction for Nonlinear Systems](https://mlsanity.com/paper/2110.00731/) (2021)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [Automated and Sound Synthesis of Lyapunov Functions with SMT Solvers](https://mlsanity.com/paper/2007.10865/) (2020)
- [Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation](https://mlsanity.com/paper/2112.14415/) (2021)
- [Towards Learning and Verifying Maximal Neural Lyapunov Functions](https://mlsanity.com/paper/2304.07215/) (2023)
- [Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition](https://mlsanity.com/paper/2001.08423/) (2020)
- [Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications](https://mlsanity.com/paper/2009.04432/) (2020)
- [Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems](https://mlsanity.com/paper/2103.12825/) (2021)
- [A Sum-of-Squares approach to the Stability and Control of Interconnected Systems using Vector Lyapunov Functions](https://mlsanity.com/paper/1501.05266/) (2015)

## Built on by (most cited first, 20 of 20)

- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)
- [Actor-Critic Physics-informed Neural Lyapunov Control](https://mlsanity.com/paper/2403.08448/) (2024)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Deep Learning for Computing Convergence Rates of Markov Chains](https://mlsanity.com/paper/2405.20435/) (2024)
- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (2024)
- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [Safe Domains of Attraction for Discrete-Time Nonlinear Systems: Characterization and Verifiable Neural Network Estimation](https://mlsanity.com/paper/2506.13961/) (2025)
- [Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributions](https://mlsanity.com/paper/2508.16737/) (2025)
- [Off Policy Lyapunov Stability in Reinforcement Learning](https://mlsanity.com/paper/2509.09863/) (2025)
- [Manifold-Guided Lyapunov Control with Diffusion Models](https://mlsanity.com/paper/2403.17692/) (2024)
- [Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems](https://mlsanity.com/paper/2412.02807/) (2024)
- [Neural Control Barrier Functions from Physics Informed Neural Networks](https://mlsanity.com/paper/2504.11045/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [(Un)supervised Learning of Maximal Lyapunov Functions](https://mlsanity.com/paper/2408.17246/) (2024)
- [Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness](https://mlsanity.com/paper/2509.21879/) (2025)
- [Physics-Informed Machine Learning for Characterizing System Stability](https://mlsanity.com/paper/2511.08831/) (2025)
- [Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation](https://mlsanity.com/paper/2603.03082/) (2026)
- [Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees](https://mlsanity.com/paper/2603.19165/) (2026)
- [Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/2603.19545/) (2026)
- [Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows](https://mlsanity.com/paper/2607.14272/) (2026)

## Abstract (arXiv)

We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov functions. We analyze the analytical properties of the solutions to the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov equation in training neural Lyapunov functions can lead to approximate regions of attraction close to the true domain of attraction. We also examine approximation errors and the convergence of neural approximations to the unique solution of Zubov's equation. We then provide sufficient conditions for the learned neural Lyapunov functions that can be readily verified by satisfiability modulo theories (SMT) solvers, enabling formal verification of both local stability analysis and region-of-attraction estimates in the large. Through a number of nonlinear examples, ranging from low to high dimensions, we demonstrate that the proposed framework can outperform traditional sums-of-squares (SOS) Lyapunov functions obtained using semidefinite programming (SDP).