# Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition

arXiv: https://arxiv.org/abs/2001.08423 · HTML: https://mlsanity.com/paper/2001.08423/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2020.

## Built on by (most cited first, 4 of 4)

- [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)
- [F-INR: Functional Tensor Decomposition for Implicit Neural Representations](https://mlsanity.com/paper/2503.21507/) (2025)
- [Lyapunov Stability Learning with Nonlinear Control via Inductive Biases](https://mlsanity.com/paper/2511.01283/) (2025)

## Abstract (arXiv)

We propose a deep neural network architecture for storing approximate Lyapunov functions of systems of ordinary differential equations. Under a small-gain condition on the system, the number of neurons needed for an approximation of a Lyapunov function with fixed accuracy grows only polynomially in the state dimension, i.e., the proposed approach is able to overcome the curse of dimensionality.