2026-09-01 04:00 UTCOriginal source2 min readUpdated: 2026-09-01 09:05 UTC

Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

arXiv:2608.28853v1 Announce Type: new 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.

SourcearXiv Machine LearningAuthor: Alessio Borgi, Mario Severino, Fabrizio Silvestri, Pietro Li\`o

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[Submitted on 28 Aug 2026]

Title:Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

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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.

Subjects:

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

Cite as: arXiv:2608.28853 [cs.LG]

(or arXiv:2608.28853v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.28853

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Alessio Borgi Dr. [view email] [v1] Fri, 28 Aug 2026 20:49:43 UTC (1,189 KB)

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