arXiv：cs.LG（机器学习，全量分类）· Alessio Borgi, Mario Severino, Fabrizio Silvestri, Pietro Li\`o·· 15 小时前AI 评分34
ESNN：在图上学习几何传输的等变层神经网络
Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs
AI 导读
研究者提出等变层神经网络 ESNN，通过在相邻向量特征间学习有向矩阵值传输来增强几何交互，同时保持精确的欧几里得等变性。该方法不提升表示阶数，而是将几何灵活性放在边传输中，并证明当相对位移是唯一协变几何输入时，每个线性 O(n) 等变映射可分解为独立的径向与切向分量。
正文
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:2608.28853v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.28853 arXiv-issued DOI via DataCite |
Submission history
From: Alessio Borgi Dr. [view email]
[v1]
Fri, 28 Aug 2026 20:49:43 UTC (1,189 KB)
[v2]
Wed, 30 Sep 2026 14:37:14 UTC (2,581 KB)
[v3]
Thu, 1 Oct 2026 11:48:09 UTC (2,576 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org