# Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networks

arXiv: https://arxiv.org/abs/2305.18965 · HTML: https://mlsanity.com/paper/2305.18965/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 72 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 30 of 46)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)

## Built on by (most cited first, 12 of 12)

- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approach](https://mlsanity.com/paper/2310.06396/) (2023)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [PANDA: Expanded Width-Aware Message Passing Beyond Rewiring](https://mlsanity.com/paper/2406.03671/) (2024)
- [FedSheafHN: Personalized Federated Learning on Graph-structured Data](https://mlsanity.com/paper/2405.16056/) (2024)
- [Graph Neural Networks with a Distribution of Parametrized Graphs](https://mlsanity.com/paper/2310.16401/) (2023)
- [Symplectic Structure-Aware Hamiltonian (Graph) Embeddings](https://mlsanity.com/paper/2309.04885/) (2023)
- [Information propagation dynamics in Deep Graph Networks](https://mlsanity.com/paper/2410.10464/) (2024)
- [Graph Spring Neural ODEs for Link Sign Prediction](https://mlsanity.com/paper/2412.12916/) (2024)
- [Simple Graph Contrastive Learning via Fractional-order Neural Diffusion Networks](https://mlsanity.com/paper/2504.16748/) (2025)
- [Adaptive Multi-view Graph Contrastive Learning via Fractional-order Neural Diffusion Networks](https://mlsanity.com/paper/2511.06216/) (2025)

## Abstract (arXiv)

In the graph node embedding problem, embedding spaces can vary significantly for different data types, leading to the need for different GNN model types. In this paper, we model the embedding update of a node feature as a Hamiltonian orbit over time. Since the Hamiltonian orbits generalize the exponential maps, this approach allows us to learn the underlying manifold of the graph in training, in contrast to most of the existing literature that assumes a fixed graph embedding manifold with a closed exponential map solution. Our proposed node embedding strategy can automatically learn, without extensive tuning, the underlying geometry of any given graph dataset even if it has diverse geometries. We test Hamiltonian functions of different forms and verify the performance of our approach on two graph node embedding downstream tasks: node classification and link prediction. Numerical experiments demonstrate that our approach adapts better to different types of graph datasets than popular state-of-the-art graph node embedding GNNs. The code is available at urlhttps://github.com/zknus/Hamiltonian-GNN.