Efficient continuous spatio-temporal physics simulation with graph spline networks

Abstract

Complex simulation of physical systems is an invaluable tool for an increasing number of fields, including engineering and scientific computing. To overcome the computational requirements of high–accuracy solvers, learned graph neural network simulators have recently been introduced. However, these methods often require a large number of nodes and edges, which can hinder their performance. Moreover, they cannot evaluate continuous solutions in space and time due to their inherently discretized structure. In this paper, we propose GraphSplineNets, a novel method to exploit the synergy between graph neural networks and orthogonal spline collocation (OSC) to accelerate learned simulations of physical systems by interpolating solutions of graph neural networks. First, we employ an encoder-decoder message passing graph neural network to map the location and value of nodes from the physical domain to hidden space and learn to predict future values. Then, to realize fully continuous simulations over the domain without dense sampling of nodes, we post–process predictions with OSC. This strategy allows us to produce a solution at any location in space and time without explicit prior knowledge of underlying differential equations and with a lower computational burden compared to learned graph simulators. We evaluate the performance of our approach in heat equation, dam breaking, and flag simulations with different graph neural network baselines, where we show consistent Pareto efficiency improvements in terms of simulation accuracy and inference time.