Isometric Transformation Invariant and Equivariant Graph Convolutional Networks

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 object detection, structural chemistry analyses, and physical simulation. A crucial requirement to applying a graph in a Euclidean space is learning the isometric transformation invariant and equivariant features. In the present paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks (GCNs), called IsoGCNs. We demonstrate that the proposed model outperforms state-of-the-art methods on tasks related with geometrical and physical data. Moreover, the proposed model can scale up to the graphs with 1M vertices and conduct an inference faster than a conventional finite element analysis. Graph-structured data embedded in a Euclidean space can be utilized in many different fields such as object detection, structural chemistry analysis, and physical simulation. Graph neural networks (GNNs) have been introduced to deal with such data. Crucial properties of a GNN include its permutation invariance and equivariance.