  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.
