ESNN Enhances Equivariant GNNs with Geometric Transport.

Equivariant graph neural networks (GNNs) are crucial for modeling systems with inherent geometric symmetries, but current first-order architectures have limitations in how vector information is transformed across a graph. This research introduces Equivariant Sheaf Neural Networks (ESNN), a novel architecture that enriches this interaction by learning directed, matrix-valued transport mechanisms between neighboring vector features. Crucially, ESNN maintains exact Euclidean equivariance. Unlike approaches that increase the order of representation, ESNN keeps scalar and vector features first-order, instead embedding additional geometric flexibility within the edge transport itself. The theoretical analysis shows that linear O(n)-equivariant maps decompose into independent radial and tangential components when relative displacement is the sole covariant input, while learned covariant features enable more complex, feature-conditioned transformations. The paper also introduces controlled symmetry relaxation for systems with a preferred ambient direction, which can be either predefined or inferred from data. ESNN has been tested across a diverse range of applications, including particle dynamics, mesh-based simulation, point-cloud classification, and molecular property prediction. The results consistently show improvements in dynamics prediction, successful recovery of gravity axes when symmetry is broken, substantial gains in selected mesh tasks and long-horizon rollouts, and robust performance against unseen rotations. This demonstrates that learning how geometric information is transported across edges provides a powerful and complementary path to more expressive equivariant message passing, without the computational burden of higher-order representations.