# SympGNNs: Symplectic Graph Neural Networks for identifiying high-dimensional Hamiltonian systems and node classification

arXiv: https://arxiv.org/abs/2408.16698 · HTML: https://mlsanity.com/paper/2408.16698/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 29 Aug 2024.

## Builds on (most cited first, 24 of 24)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [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)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [Is Homophily a Necessity for Graph Neural Networks?](https://mlsanity.com/paper/2106.06134/) (2021)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Permutation Equivariant Graph Framelets for Heterophilous Graph Learning](https://mlsanity.com/paper/2306.04265/) (2023)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [HAQJSK: Hierarchical-Aligned Quantum Jensen-Shannon Kernels for Graph Classification](https://mlsanity.com/paper/2211.02904/) (2022)

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

- [Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs](https://mlsanity.com/paper/2505.03595/) (2025)
- [Rapid training of Hamiltonian graph networks using random features](https://mlsanity.com/paper/2506.06558/) (2025)
- [Learning Nonholonomic Dynamics with Constraint Discovery](https://mlsanity.com/paper/2410.15201/) (2024)
- [Noise-robust Contrastive Learning for Critical Transition Detection in Dynamical Systems](https://mlsanity.com/paper/2512.12523/) (2025)

## Abstract (arXiv)

Existing neural network models to learn Hamiltonian systems, such as SympNets, although accurate in low-dimensions, struggle to learn the correct dynamics for high-dimensional many-body systems. Herein, we introduce Symplectic Graph Neural Networks (SympGNNs) that can effectively handle system identification in high-dimensional Hamiltonian systems, as well as node classification. SympGNNs combines symplectic maps with permutation equivariance, a property of graph neural networks. Specifically, we propose two variants of SympGNNs: i) G-SympGNN and ii) LA-SympGNN, arising from different parameterizations of the kinetic and potential energy. We demonstrate the capabilities of SympGNN on two physical examples: a 40-particle coupled Harmonic oscillator, and a 2000-particle molecular dynamics simulation in a two-dimensional Lennard-Jones potential. Furthermore, we demonstrate the performance of SympGNN in the node classification task, achieving accuracy comparable to the state-of-the-art. We also empirically show that SympGNN can overcome the oversmoothing and heterophily problems, two key challenges in the field of graph neural networks.