# Learning Region of Attraction for Nonlinear Systems

arXiv: https://arxiv.org/abs/2110.00731 · HTML: https://mlsanity.com/paper/2110.00731/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 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 Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Neural Lyapunov Control for Discrete-Time Systems](https://mlsanity.com/paper/2305.06547/) (2023)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Verification-Aided Learning of Neural Network Barrier Functions with Termination Guarantees](https://mlsanity.com/paper/2403.07308/) (2024)
- [Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systems](https://mlsanity.com/paper/2210.08339/) (2022)
- [Safe Domains of Attraction for Discrete-Time Nonlinear Systems: Characterization and Verifiable Neural Network Estimation](https://mlsanity.com/paper/2506.13961/) (2025)
- [Control invariant set enhanced safe reinforcement learning: improved sampling efficiency, guaranteed stability and robustness](https://mlsanity.com/paper/2305.15602/) (2023)
- [Model-free Learning of Regions of Attraction via Recurrent Sets](https://mlsanity.com/paper/2204.10372/) (2022)
- [A Policy Optimization Method Towards Optimal-time Stability](https://mlsanity.com/paper/2301.00521/) (2023)
- [Control invariant set enhanced reinforcement learning for process control: improved sampling efficiency and guaranteed stability](https://mlsanity.com/paper/2304.05509/) (2023)
- [Neural Lyapunov Function Approximation with Self-Supervised Reinforcement Learning](https://mlsanity.com/paper/2503.15629/) (2025)
- [Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control](https://mlsanity.com/paper/2201.00801/) (2022)
- [Optimizing the switching operation in monoclonal antibody production: Economic MPC and reinforcement learning](https://mlsanity.com/paper/2308.03928/) (2023)
- [V-MORALS: Visual Morse Graph-Aided Estimation of Regions of Attraction in a Learned Latent Space](https://mlsanity.com/paper/2602.23524/) (2026)
- [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)

## Abstract (arXiv)

Estimating the region of attraction (ROA) of general nonlinear autonomous systems remains a challenging problem and requires a case-by-case analysis. Leveraging the universal approximation property of neural networks, in this paper, we propose a counterexample-guided method to estimate the ROA of general nonlinear dynamical systems provided that they can be approximated by piecewise linear neural networks and that the approximation error can be bounded. Specifically, our method searches for robust Lyapunov functions using counterexamples, i.e., the states at which the Lyapunov conditions fail. We generate the counterexamples using Mixed-Integer Quadratic Programming. Our method is guaranteed to find a robust Lyapunov function in the parameterized function class, if exists, after collecting a finite number of counterexamples. We illustrate our method through numerical examples.