Symplectic Structure-Aware Hamiltonian (Graph) Embeddings

TL;DR

SAH-GNN introduces a symplectic structure-aware Hamiltonian framework for graph neural networks, enabling adaptive, energy-conserving node embeddings that improve performance across diverse datasets.

Contribution

It generalizes Hamiltonian GNNs by employing Riemannian optimization on the symplectic Stiefel manifold, allowing automatic adaptation to various graph geometries.

Findings

Outperforms existing Hamiltonian GNNs in node classification tasks

Automatically adapts to different graph datasets without extensive tuning

Conserves energy, ensuring physically meaningful embeddings

Abstract

In traditional Graph Neural Networks (GNNs), the assumption of a fixed embedding manifold often limits their adaptability to diverse graph geometries. Recently, Hamiltonian system-inspired GNNs have been proposed to address the dynamic nature of such embeddings by incorporating physical laws into node feature updates. We present Symplectic Structure-Aware Hamiltonian GNN (SAH-GNN), a novel approach that generalizes Hamiltonian dynamics for more flexible node feature updates. Unlike existing Hamiltonian approaches, SAH-GNN employs Riemannian optimization on the symplectic Stiefel manifold to adaptively learn the underlying symplectic structure, circumventing the limitations of existing Hamiltonian GNNs that rely on a pre-defined form of standard symplectic structure. This innovation allows SAH-GNN to automatically adapt to various graph datasets without extensive hyperparameter tuning. Moreover, it conserves energy during training meaning the implicit Hamiltonian system is physically meaningful. Finally, we empirically validate SAH-GNN's superiority and adaptability in node classification tasks across multiple types of graph datasets.