GRAPH SPLINE NETWORKS FOR EFFICIENT CONTINU-OUS 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 OSC (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 GRAPHSPLINENETS in predicting complex dynamical systems such as the heat equation, damped wave propagation and the Navier-Stokes equations for incompressible flows, where they improve accuracy of more than 25% while providing at least 60% speedup.

1. INTRODUCTION

For a growing variety of fields, simulations of partial differential equations (PDEs) representing physical processes are an essential tool. PDE-based simulators have been widely employed in a range of practical issues, spanning from astrophysics (Mücke et al., 2000) to biology (Quarteroni & Veneziani, 2003 ), engineering (Wu & Porté-Agel, 2011 ), finance, (Marriott et al., 2015) or weather forecasting (Bauer et al., 2015) . Traditional solvers for phsysics-based simulation oftentimes need a significant amount of computational resources (Houska et al., 2012) , such as solvers based on first principles and the modified Gauss-Newton methods. To broaden the scope of applications of dynamics simulation, the scientific machine learning community has put considerable effort into developing computationally simple yet accurate simulation approaches. Deep learning has been shown to be a powerful alternative to efficiently compute solutions (Raissi et al., 2019) or model dynamical systems directly from data (Mrowca et al., 2018) . Among deep learning methods, graph neural networks (GNNs) come with desirable properties such as spatial equivariance and translational invariance which allow learning representations of dynamical interactions in a generalizable manner (Pfaff et al., 2021; Bronstein et al., 2021) and on unstructured grids. Despite the benefits of these paradigms, graph-based models have the fundamental drawback of being discrete in nature, which makes it challenging to implement continuous simulations in time and space. While graph models that operate in continuous space or continuous time have been introduced in the past (Poli et al., 2019) , such approaches mainly deal with only one aspect of continuity at once, either in space or time, and are hindered by accuracy issues while interpolating in space (Alet et al., 2019) or require a considerable number of iterative evaluations of a vector field in time limiting their performance (Xhonneux et al., 2020) . To bridge the gap between the inherently discrete graphs and the intrinsic continuous nature of the real world, in this work we propose GRAPHSPLINENETS, a novel method that exploits the synergy between graph neural networks and the orthogonal spline collocation (OSC) method (Bialecki & Fairweather, 2001; Fairweather & Meade, 2020) . By leveraging the OSC, our approach can

