  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.
