Title: Graph Spline Networks for Efficient Continuous Simulation of Dynamical Systems. Abstract: While complex simulations of physical systems have been widely studied in engineering and scientific computing, lowering their often prohibitive computational requirements has only recently been tackled by deep learning approaches. In this paper, we present GraphSplineNets, a novel deep learning approach to speed up simulation of physical systems with spatio-temporal continuous outputs by exploiting the synergy between graph neural networks (GNN) and orthogonal spline collocation (OSC). Two differentiable time-oriented OSC and spatial-oriented OSC are applied to bridge the gap between discrete GNN outputs and generate continuous solutions at any location in space and time without explicit prior knowledge of underlying differential equations. Moreover, we introduce an adaptive collocation strategy in space to enable the model to sample from the most important regions. Our model improves on widely used graph neural networks for physics simulation on both efficiency and solution accuracy. We demonstrate SplineGraphNets in predicting complex dynamical systems such as the heat equation, damped wave propagation and the Navier-Stokes equations for incompressible flow, where they improve accuracy of more than 25% while providing at least 60% speedup. 