Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

Equivariant graph neural networks constrain how geometric features transform, but efficient first-order architectures remain limited in how vector information can be transported across graph edges. We introduce ESNN, an Equivariant Sheaf Neural Network that learns directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance. Rather than increasing representation order, ESNN retains first-order scalar and vector features and places additional geometric expressivity in the edge interactions themselves. Its transport mechanisms range from isotropic transformations to feature-conditioned rotations and anisotropic radial–tangential actions, allowing vector messages to be transformed according to the local geometry before aggregation. We also equip ESNN with a controlled symmetry-relaxation mechanism for systems with a preferred ambient direction, preserving exact equivariance to the remaining symmetry while recovering full $E(n)$-equivariance as a special case. Across diverse geometric learning problems spanning physical dynamics, point clouds, and molecular systems, ESNN improves predictive performance over first-order equivariant baselines, learns a latent preferred direction from data, and remains robust to unseen rotations. These results establish edge-wise geometric transport as a source of equivariant expressivity that does not require higher-order representations.