mlsanityData to 14 Oct 2026

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

arXiv 2112.14415 ↗First posted 29 Dec 2021math.DS cs.LG cs.SYcited by 7

Not on the map. Why?

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.

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

13 of 53 references.

References · 13

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1 more
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Cited by year

Citing papers per year: 2023: 1; 2024: 3; 2025: 3.

7 papers.

  1. 2023Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verificationcited by 20
  2. 2024LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attractioncited by 10
  3. 2024Actor-Critic Physics-informed Neural Lyapunov Controlcited by 8
  4. 2025Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansioncited by 6
  5. 2024Formally Verified Physics-Informed Neural Control Lyapunov Functionscited by 4
  6. 2025Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustnesscited by 0
  7. 2025Physics-Informed Machine Learning for Characterizing System Stabilitycited by 0