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.

翻译：传统图神经网络（GNNs）中固定嵌入流形的假设常常限制了其对多样化图几何结构的适应性。近期，哈密顿系统启发的GNNs通过将物理定律融入节点特征更新，被提出以应对此类嵌入的动态特性。我们提出辛结构感知哈密顿图神经网络（SAH-GNN），这是一种泛化哈密顿动力学以实现更灵活节点特征更新的创新方法。与现有哈密顿方法不同，SAH-GNN在辛施蒂费尔流形上采用黎曼优化来自适应学习底层辛结构，规避了依赖预定义标准辛结构形式的现有哈密顿GNN的局限性。这一创新使得SAH-GNN无需大量超参数调优即可自动适应各类图数据集。此外，该方法在训练过程中保持能量守恒，意味着隐式哈密顿系统具有物理意义。最终，我们在多种图数据集的节点分类任务中实证验证了SAH-GNN的优越性与适应性。