Symplectic Structure-Aware Hamiltonian (Graph) Embeddings
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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 are proposed to address the dynamic nature of such embeddings by incorporating physical laws into node feature updates. In this work, we present SAH-GNN, a novel approach that generalizes Hamiltonian dynamics for more flexible node feature updates. Unlike existing Hamiltonian-inspired GNNs, SAH-GNN employs Riemannian optimization on the symplectic Stiefel manifold to adaptively learn the underlying symplectic structure during training, 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 such that the implicit Hamiltonian system is physically meaningful. To this end, we empirically validate SAH-GNN's superior performance and adaptability in node classification tasks across multiple types of graph datasets.
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