# The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems

arXiv: https://arxiv.org/abs/1808.00924 · HTML: https://mlsanity.com/paper/1808.00924/

## Key facts

- Cited by 88 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2018.

## Built on by (most cited first, 30 of 88)

- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [How to Certify Machine Learning Based Safety-critical Systems? A Systematic Literature Review](https://mlsanity.com/paper/2107.12045/) (2021)
- [Learning Barrier Certificates: Towards Safe Reinforcement Learning with Zero Training-time Violations](https://mlsanity.com/paper/2108.01846/) (2021)
- [Trustworthy Reinforcement Learning Against Intrinsic Vulnerabilities: Robustness, Safety, and Generalizability](https://mlsanity.com/paper/2209.08025/) (2022)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Lyapunov Density Models: Constraining Distribution Shift in Learning-Based Control](https://mlsanity.com/paper/2206.10524/) (2022)
- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [Neural Lyapunov Control for Discrete-Time Systems](https://mlsanity.com/paper/2305.06547/) (2023)
- [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023)
- [Value Iteration in Continuous Actions, States and Time](https://mlsanity.com/paper/2105.04682/) (2021)
- [Learning Stable Deep Dynamics Models for Partially Observed or Delayed Dynamical Systems](https://mlsanity.com/paper/2110.14296/) (2021)
- [Control-oriented meta-learning](https://mlsanity.com/paper/2204.06716/) (2022)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates](https://mlsanity.com/paper/2405.14058/) (2024)
- [How Generalizable Is My Behavior Cloning Policy? A Statistical Approach to Trustworthy Performance Evaluation](https://mlsanity.com/paper/2405.05439/) (2024)
- [Scalable Learning of Safety Guarantees for Autonomous Systems using Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2101.05916/) (2021)
- [Compositional Policy Learning in Stochastic Control Systems with Formal Guarantees](https://mlsanity.com/paper/2312.01456/) (2023)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach](https://mlsanity.com/paper/2112.09046/) (2021)
- [Neural Koopman Lyapunov Control](https://mlsanity.com/paper/2201.05098/) (2022)
- [LS3: Latent Space Safe Sets for Long-Horizon Visuomotor Control of Sparse Reward Iterative Tasks](https://mlsanity.com/paper/2107.04775/) (2021)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)
- [Robust Value Iteration for Continuous Control Tasks](https://mlsanity.com/paper/2105.12189/) (2021)
- [RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks](https://mlsanity.com/paper/2106.08928/) (2021)

## Abstract (arXiv)

Learning algorithms have shown considerable prowess in simulation by allowing robots to adapt to uncertain environments and improve their performance. However, such algorithms are rarely used in practice on safety-critical systems, since the learned policy typically does not yield any safety guarantees. That is, the required exploration may cause physical harm to the robot or its environment. In this paper, we present a method to learn accurate safety certificates for nonlinear, closed-loop dynamical systems. Specifically, we construct a neural network Lyapunov function and a training algorithm that adapts it to the shape of the largest safe region in the state space. The algorithm relies only on knowledge of inputs and outputs of the dynamics, rather than on any specific model structure. We demonstrate our method by learning the safe region of attraction for a simulated inverted pendulum. Furthermore, we discuss how our method can be used in safe learning algorithms together with statistical models of dynamical systems.