  While complex simulations of physical systems have been widely used 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
method to speed up the forecasting of physical systems by reducing the grid
size and number of iteration steps of deep surrogate models. Our method uses
two differentiable orthogonal spline collocation methods to efficiently predict
response at any location in time and space. Additionally, we introduce an
adaptive collocation strategy in space to prioritize sampling from the most
important regions. GraphSplineNets improve the accuracy-speedup tradeoff in
forecasting various dynamical systems with increasing complexity, including the
heat equation, damped wave propagation, Navier-Stokes equations, and real-world
ocean currents in both regular and irregular domains.
