发表机构
University of Cambridge; Sapienza University of Rome; University of Padua（剑桥大学; 罗马第一大学; 帕多瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出等变层状神经网络（ESNN），通过学习图上边的矩阵值几何传输，在保持等变性的同时提升了多个几何相关任务的性能。
AI 中文摘要
等变图神经网络为几何系统建模提供了原理性方法，但高效的一阶架构在向量信息跨图传播时的变换方式仍存在局限。我们提出\textsc{ESNN}（等变层状神经网络），该模型通过学习相邻向量特征间有向的矩阵值传输来丰富这种交互，同时保持精确的欧几里得等变性。ESNN不提升表示的阶数，而是将标量和向量特征保持为一阶，并将额外的几何灵活性置于边传输本身。我们从理论上对该传输进行了表征，表明当相对位移是唯一的协变几何输入时，每个线性$O(n)$等变映射可分解为独立的径向和切向分量，而学习到的协变特征可实现更丰富的特征条件变换。我们还针对具有偏好环境方向的系统引入了可控对称性松弛，该方向可被指定或从数据中推断，且当方向通路失活时可恢复完整的$E(n)$等变性。在粒子动力学、基于网格的模拟、点云分类和分子性质预测任务中，ESNN提升了动力学预测性能，在对称性被破坏时能恢复重力轴，在选定的网格任务和长时程滚动中取得了显著增益，且对未见过的旋转保持鲁棒性。这些结果表明，学习几何信息如何跨边传输为表达性等变消息传递提供了一条互补路径，无需高阶表示。
英文摘要
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.
