BI-STRIDE MULTI-SCALE GRAPH NEURAL NETWORK FOR MESH-BASED PHYSICAL SIMULATION

Abstract

Learning physical systems on unstructured meshes by flat Graph neural networks (GNNs) faces the challenge of modeling the long-range interactions due to the scaling complexity w.r.t. the number of nodes, limiting the generalization under mesh refinement. On regular grids, the convolutional neural networks (CNNs) with a U-net structure can resolve this challenge by efficient stride, pooling, and upsampling operations. Nonetheless, these tools are much less developed for graph neural networks (GNNs), especially when GNNs are employed for learning large-scale mesh-based physics. The challenges arise from the highly irregular meshes and the lack of effective ways to construct the multi-level structure without losing connectivity. Inspired by the bipartite graph determination algorithm, we introduce Bi-Stride Multi-Scale Graph Neural Network (BSMS-GNN) by proposing bi-stride as a simple pooling strategy for building the multi-level GNN. Bi-stride pools nodes by striding every other Breadth-First-Search (BFS) frontier; it 1) works robustly on any challenging mesh in the wild, 2) avoids using a mesh generator at coarser levels, 3) avoids the spatial proximity for building coarser levels, and 4) uses non-parametrized aggregating/returning instead of MLPs during pooling and unpooling. Experiments show that our framework significantly outperforms the state-of-the-art method's computational efficiency in representative physics-based simulation cases.

1. INTRODUCTION

Simulating physical systems through numerically solving partial differential equations (PDEs) plays a key role in various science and engineering applications, ranging from particle-based (Jiang et al., 2016) and mesh-based (Li et al., 2020a) solid mechanics to grid-based fluid (Bridson, 2015) and aero (Cao et al., 2022) dynamics. Despite the extensive successes in improving their stability, accuracy, and efficiency, numerical solvers are often computationally expensive for time-sensitive applications, especially iterative design optimization where fast online inferring is desired. Recently, machine learning approaches have demonstrated impressive potential in improving the efficiency of inferring physical states with competitive accuracy. Representative methods include end-to-end frameworks (Obiols-Sales et al., 2020) and those with physics-informed neural networks (PINNs) (Raissi et al., 2019; Karniadakis et al., 2021; Sun et al., 2020; Gao et al., 2021) . Many existing works apply convolutional neural networks (CNNs) (Fukushima & Miyake, 1982) to learn physical systems on two-or three-dimensional structured grids (Kim et al., 2019; Fotiadis et al., 2020; Gao et al., 2021; Guo et al., 2016; Tompson et al., 2017) . It is generally recognized that CNNs exhibit strong performance on handling local information with convolution and global information with pooling/upsampling. However, the strict dependency on regular domain shapes makes it non-trivial to be applied on unstructured meshes. Although it is possible to deform the domains to rectangular shapes to apply CNNs (Gao et al., 2021) or other models, such as NeuralOpera-torNets (Li et al., 2022) , the challenge remains for domains with complex topologies, which are common in practice. On the other hand, graph neural networks (GNNs) have been considered as a natural choice for physics-based simulation on unstructured meshes (Battaglia et al., 2018; Belbute-Peres et al., 2020; Gao et al., 2022; Harsch & Riedelbauch, 2021; Pfaff et al., 2020; Sanchez-Gonzalez et al., 2018; 2020) . However, all the above methods use the flat GNN that faces two challenges when the graph

