# Lyapunov Stable Graph Neural Flow

arXiv: https://arxiv.org/abs/2603.12557 · HTML: https://mlsanity.com/paper/2603.12557/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 52 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 30 of 30)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Graph Structure Learning for Robust Graph Neural Networks](https://mlsanity.com/paper/2005.10203/) (2020)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Graph neural networks for materials science and chemistry](https://mlsanity.com/paper/2208.09481/) (2022)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [TDGIA:Effective Injection Attacks on Graph Neural Networks](https://mlsanity.com/paper/2106.06663/) (2021)
- [Understanding and Improving Graph Injection Attack by Promoting Unnoticeability](https://mlsanity.com/paper/2202.08057/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Graph Robustness Benchmark: Benchmarking the Adversarial Robustness of Graph Machine Learning](https://mlsanity.com/paper/2111.04314/) (2021)
- [On the Robustness of Graph Neural Diffusion to Topology Perturbations](https://mlsanity.com/paper/2209.07754/) (2022)
- [Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approach](https://mlsanity.com/paper/2310.06396/) (2023)
- [A survey of dynamic graph neural networks](https://mlsanity.com/paper/2404.18211/) (2024)
- [Learning Dynamics Models with Stable Invariant Sets](https://mlsanity.com/paper/2006.08935/) (2020)
- [Learning Stable Deep Dynamics Models for Partially Observed or Delayed Dynamical Systems](https://mlsanity.com/paper/2110.14296/) (2021)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Modern Methods in Associative Memory](https://mlsanity.com/paper/2507.06211/) (2025)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [Finite-time Lyapunov exponents of deep neural networks](https://mlsanity.com/paper/2306.12548/) (2023)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)
- [Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study](https://mlsanity.com/paper/2401.04331/) (2024)
- [Graph Neural Networks for Graph Drawing](https://mlsanity.com/paper/2109.10061/) (2021)
- [Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structure](https://mlsanity.com/paper/2401.04986/) (2024)
- [Neural Variable-Order Fractional Differential Equation Networks](https://mlsanity.com/paper/2503.16207/) (2025)

## Abstract (arXiv)

Graph Neural Networks (GNNs) are highly vulnerable to adversarial perturbations in both topology and features, making the learning of robust representations a critical challenge. In this work, we bridge GNNs with control theory to introduce a novel defense framework grounded in integer- and fractional-order Lyapunov stability. Unlike conventional strategies that rely on resource-heavy adversarial training or data purification, our approach fundamentally constrains the underlying feature-update dynamics of the GNN. We propose an adaptive, learnable Lyapunov function paired with a novel projection mechanism that maps the network's state into a stable space, thereby offering theoretically provable stability guarantees. Notably, this mechanism is orthogonal to existing defenses, allowing for seamless integration with techniques like adversarial training to achieve cumulative robustness. Extensive experiments demonstrate that our Lyapunov-stable graph neural flows substantially outperform base neural flows and state-of-the-art baselines across standard benchmarks and various adversarial attack scenarios.