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

TL;DR

The paper tackles the challenge of learning high-dimensional Hamiltonian dynamics from limited data by introducing SympGNNs, which fuse symplectic structure with permutation-equivariant graph networks. It presents two variants, G-SympGNN and LA-SympGNN, that preserve the symplectic flow while exploiting graph structure to model many-body interactions, enabling both system identification and node classification. Empirically, the approach demonstrates superior performance in data-scarce regimes on a 40-particle coupled oscillator and a 2000-particle Lennard-Jones MD, while also delivering strong results on benchmark node-classification tasks and mitigating oversmoothing and heterophily. Overall, SympGNNs offer a principled, structure-preserving framework for scalable physics-informed graph learning with practical impact on both dynamical-system identification and graph-based inference.

Abstract

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.

Figures (10)

• Figure 1: Commutative diagram for a symplectically permutation equivariant map. The SympGNN update from timestep $k$ to $k+1$ followed by a permutation, yields the same result as a permutation followed by SympGNN update, demonstrating that the map is equivariant to permutations.
• Figure 2: Illustration of the low and up modules in G-SympGNN. This figure shows the update rules from timestep $k$ to $k+1$ for the low module, which updates $\boldsymbol{q}$, and the up module, which updates $\boldsymbol{p}$. The G-SympGNN is constructed by alternating these low and up modules.
• Figure 3: Illustration of the linear low layer and the activation low layer in LA-SympGNN for $\square = A$. The linear up layer updates $\boldsymbol{q}$ based on its neigbhors through message passing.
• Figure 4: Illustrative overview of the simulation results. We present results on system identification in two physical systems with Hamiltonian dynamics: i) coupled harmonic oscillator and ii) 2D Lennard-Jones particles. We also present results on node classification in i) four benchmark datasets, ii) the oversmoothing problem and iii) the heterophily problem.
• Figure 5: Comparison of SympNet vs SympGNN when the training set size $T = 500$. We randomly sample three particles from the chain and plot the trajectory in the test window. SympGNN outperforms SympNet in matching the ground truth trajectory when provided limited training data. This is because more inductive biases (permutation equivariance) is embedded.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Remark 1
• Remark 2
• Definition 5: G-SympGNN
• Theorem 2
• Definition 6: LA-SympGNN