Wallpaper Net: A general Blueprint for Wallpaper-Group-Equivariant Graph Neural Networks

Equivariant graph neural networks are typically designed for continuous Euclidean symmetry groups such as $E(3)$ or $SE(3)$, where arbitrary global translations are symmetries and can be removed using relative coordinates. In many settings, however, a graph is defined relative to an underlying periodic structure, so its symmetry is no longer the full Euclidean group. In two dimensions, these symmetries are the 17 wallpaper groups. This structure creates an opportunity: rather than explicitly representing the entire periodic system as a large graph, one can represent only the smaller graph of interest while incorporating the symmetry of the surrounding periodic structure into the model. We present a general blueprint for wallpaper-group-equivariant message passing by separating the symmetry problem into two components: point-group equivariance and lattice-relative translational symmetry. We demonstrate how it can be applied to a molecular adsorption problem on a periodic graphene substrate, providing high accuracy at a fraction of the cost of full-system message passing.