# Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation

arXiv: https://arxiv.org/abs/2112.14415 · HTML: https://mlsanity.com/paper/2112.14415/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 29 Dec 2021.

## Builds on (most cited first, 13 of 13)

- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Computing Lyapunov functions using deep neural networks](https://mlsanity.com/paper/2005.08965/) (2020)
- [Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations](https://mlsanity.com/paper/1507.04769/) (2015)
- [Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs](https://mlsanity.com/paper/1702.04400/) (2017)
- [QRnet: optimal regulator design with LQR-augmented neural networks](https://mlsanity.com/paper/2009.05686/) (2020)
- [Algorithms of Data Development For Deep Learning and Feedback Design](https://mlsanity.com/paper/1912.00492/) (2019)
- [Neural Network Approximations of Compositional Functions With Applications to Dynamical Systems](https://mlsanity.com/paper/2012.01698/) (2020)

## Built on by (most cited first, 7 of 7)

- [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)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Formally Verified Physics-Informed Neural Control Lyapunov Functions](https://mlsanity.com/paper/2409.20528/) (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)

## Abstract (arXiv)

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.