Title: NEURAL HAMILTONIAN FLOWS IN GRAPH NEURAL NETWORKS. Abstract: Graph neural networks (GNNs) suffer from oversmoothing and oversquashing problems when node features are updated over too many layers. Embedding spaces can also vary significantly for different data types, leading to the need for different GNN model types. In this paper, we model the embedding of a node feature as a Hamiltonian flow over time. As in physics where Hamiltonian flow conserves the energy over time, its induced GNNs enable a more stable feature updating mechanism. Moreover, since the Hamiltonian flows are defined on a general symplectic manifold, this approach allows us to learn the underlying manifold of the graph in training, in contrast to most of the existing literature that assumes a fixed graph embedding manifold. We test Hamiltonian flows of different forms and demonstrate empirically that our approach achieves better node classification accuracy than popular state-of-the-art GNNs.