ISOMETRIC TRANSFORMATION INVARIANT AND EQUIVARIANT GRAPH CONVOLUTIONAL NETWORKS

Abstract

Graphs are one of the most important data structures for representing pairwise relations between objects. Specifically, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations. A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efficient manner. In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs. We demonstrate that the proposed model has a competitive performance compared to state-of-the-art methods on tasks related to geometrical and physical simulation data. Moreover, the proposed model can scale up to graphs with 1M vertices and conduct an inference faster than a conventional finite element analysis, which the existing equivariant models cannot achieve.

1. INTRODUCTION

Graph-structured data embedded in Euclidean spaces can be utilized in many different fields such as object detection, structural chemistry analysis, and physical simulations. Graph neural networks (GNNs) have been introduced to deal with such data. The crucial properties of GNNs include permutation invariance and equivariance. Besides permutations, isometric transformation invariance and equivariance must be addressed when considering graphs in Euclidean spaces because many properties of objects in the Euclidean space do not change under translation and rotation. Due to such invariance and equivariance, 1) the interpretation of the model is facilitated; 2) the output of the model is stabilized and predictable; and 3) the training is rendered efficient by eliminating the necessity of data augmentation as discussed in the literature (Thomas et al., 2018; Weiler et al., 2018; Fuchs et al., 2020) . Isometric transformation invariance and equivariance are inevitable, especially when applied to physical simulations, because every physical quantity and physical law is either invariant or equivariant to such a transformation. Another essential requirement for such applications is computational efficiency because the primary objective of learning a physical simulation is to replace a computationally expensive simulation method with a faster machine learning model. In the present paper, we propose IsoGCNs, a set of simple yet powerful models that provide computationally-efficient isometric transformation invariance and equivariance based on graph convolutional networks (GCNs) (Kipf & Welling, 2017) . Specifically, by simply tweaking the definition of an adjacency matrix, the proposed model can realize isometric transformation invariance. Because the proposed approach relies on graphs, it can deal with the complex shapes that are usually presented using mesh or point cloud data structures. Besides, a specific form of the IsoGCN layer can be regarded as a spatial differential operator that is essential for describing physical laws. In addition, we have shown that the proposed approach is computationally efficient in terms of processing graphs 1

