Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

2026-08-28

AI Summary

本文提出ESNN，通过学习图中相邻向量特征之间的矩阵值传输来改进几何系统的建模，同时保持欧几里得等变性。

Abstract

Equivariant graph neural networks provide a principled way to model geometric systems, but efficient first-order architectures remain limited in how vector information can be transformed as it moves across a graph. We introduce \textsc{ESNN}, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance. Rather than increasing the order of the representation, ESNN keeps scalar and vector features first-order and places the additional geometric flexibility in the edge transport itself. We characterize this transport theoretically, showing that when relative displacement is the only covariant geometric input, every linear $O(n)$-equivariant map decomposes into independent radial and tangential components, while learned covariant features enable richer feature-conditioned transformations. We also introduce controlled symmetry relaxation for systems with a preferred ambient direction, which may be prescribed or inferred from data while recovering full $E(n)$-equivariance when the directional pathway is inactive. Across particle dynamics, mesh-based simulation, point-cloud classification, and molecular property prediction, ESNN improves dynamics prediction, recovers the gravity axis when symmetry is broken, yields substantial gains on selected mesh tasks and long-horizon rollouts, and remains robust to unseen rotations. These results show that learning how geometric information is transported across edges offers a complementary route to expressive equivariant message passing without requiring higher-order representations.

Problem

• Equivariant Graph Neural Networks
• Geometric Transport
• Euclidean Equivariance

Innovation

• Equivariant Sheaf Neural Networks
• geometric transport
• Euclidean equivariance
• controlled symmetry relaxation