# Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

arXiv: https://arxiv.org/abs/2109.13359 · HTML: https://mlsanity.com/paper/2109.13359/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 27 Sep 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Limitations of the Lipschitz constant as a defense against adversarial examples](https://mlsanity.com/paper/1807.09705/) (2018)
- [Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions](https://mlsanity.com/paper/1807.00297/) (2018)
- [Approximating Lipschitz continuous functions with GroupSort neural networks](https://mlsanity.com/paper/2006.05254/) (2020)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [The Representation Power of Neural Networks: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2012.05451/) (2020)
- [Analytic function approximation by path norm regularized deep networks](https://mlsanity.com/paper/2104.02095/) (2021)
- [Learning-based Adaptive Control using Contraction Theory](https://mlsanity.com/paper/2103.02987/) (2021)

## Built on by (most cited first, 16 of 16)

- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [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)
- [Verification-Aided Learning of Neural Network Barrier Functions with Termination Guarantees](https://mlsanity.com/paper/2403.07308/) (2024)
- [Value Functions are Control Barrier Functions: Verification of Safe Policies using Control Theory](https://mlsanity.com/paper/2306.04026/) (2023)
- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (2024)
- [Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety](https://mlsanity.com/paper/2203.15474/) (2022)
- [Distributionally Robust Policy and Lyapunov-Certificate Learning](https://mlsanity.com/paper/2404.03017/) (2024)
- [A Policy Optimization Method Towards Optimal-time Stability](https://mlsanity.com/paper/2301.00521/) (2023)
- [Neural Lyapunov Function Approximation with Self-Supervised Reinforcement Learning](https://mlsanity.com/paper/2503.15629/) (2025)
- [Certifying Stability of Reinforcement Learning Policies using Generalized Lyapunov Functions](https://mlsanity.com/paper/2505.10947/) (2025)
- [Structured Deep Neural Network-Based Backstepping Trajectory Tracking Control for Lagrangian Systems](https://mlsanity.com/paper/2403.00381/) (2024)
- [(Un)supervised Learning of Maximal Lyapunov Functions](https://mlsanity.com/paper/2408.17246/) (2024)
- [Lyapunov Stability Learning with Nonlinear Control via Inductive Biases](https://mlsanity.com/paper/2511.01283/) (2025)
- [Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/2603.19545/) (2026)

## Abstract (arXiv)

We develop a versatile deep neural network architecture, called Lyapunov-Net, to approximate Lyapunov functions of dynamical systems in high dimensions. Lyapunov-Net guarantees positive definiteness, and thus it can be easily trained to satisfy the negative orbital derivative condition, which only renders a single term in the empirical risk function in practice. This significantly reduces the number of hyper-parameters compared to existing methods. We also provide theoretical justifications on the approximation power of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the proposed method on nonlinear dynamical systems involving up to 30-dimensional state spaces, and show that the proposed approach significantly outperforms the state-of-the-art methods.