# Graph Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/1911.07532 · HTML: https://mlsanity.com/paper/1911.07532/

## Key facts

- Cited by 154 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2019.

## Built on by (most cited first, 30 of 154)

- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [Anti-Symmetric DGN: a stable architecture for Deep Graph Networks](https://mlsanity.com/paper/2210.09789/) (2022)
- [Dissecting the Diffusion Process in Linear Graph Convolutional Networks](https://mlsanity.com/paper/2102.10739/) (2021)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Effectively Modeling Time Series with Simple Discrete State Spaces](https://mlsanity.com/paper/2303.09489/) (2023)
- [Gradient Gating for Deep Multi-Rate Learning on Graphs](https://mlsanity.com/paper/2210.00513/) (2022)
- [A Fractional Graph Laplacian Approach to Oversmoothing](https://mlsanity.com/paper/2305.13084/) (2023)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [A Review of Graph Neural Networks in Epidemic Modeling](https://mlsanity.com/paper/2403.19852/) (2024)
- [Challenges and Opportunities in Deep Reinforcement Learning with Graph Neural Networks: A Comprehensive review of Algorithms and Applications](https://mlsanity.com/paper/2206.07922/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systems](https://mlsanity.com/paper/2405.01009/) (2024)
- [Long Range Propagation on Continuous-Time Dynamic Graphs](https://mlsanity.com/paper/2406.02740/) (2024)
- [SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases](https://mlsanity.com/paper/2308.13212/) (2023)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning](https://mlsanity.com/paper/2302.06186/) (2023)
- [TIDE: Time Derivative Diffusion for Deep Learning on Graphs](https://mlsanity.com/paper/2212.02483/) (2022)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Topological Neural Networks go Persistent, Equivariant, and Continuous](https://mlsanity.com/paper/2406.03164/) (2024)

## Abstract (arXiv)

We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations. The proposed framework is shown to be compatible with various static and autoregressive GNN models. Results prove general effectiveness of GDEs: in static settings they offer computational advantages by incorporating numerical methods in their forward pass; in dynamic settings, on the other hand, they are shown to improve performance by exploiting the geometry of the underlying dynamics.