AI 中文总结
扩展几何深度学习利用更丰富对称结构，开发等变神经网络，刻画线性等变映射，构建层并证明连续等变映射的通用逼近定理，还扩展到范畴等变神经网络，给出等变神经网络通用形式及定理。
AI 中文摘要
对称性在自然和社会中无处不在。几何深度学习利用数据中的对称性来提高深度学习系统的性能和效率。在本文中，我们扩展几何深度学习以利用更丰富的对称结构。具体来说，我们开发了序等变神经网络（OENN），它通过面偏序集（面范畴）上的等变丛理论推广了标准图消息传递和层神经网络。我们（i）刻画所有线性序等变映射，（ii）构建OENN层，以及（iii）证明连续序等变映射的通用逼近定理（UATs），即使限制在层神经网络上（之前未知UAT），这些也是新结果。我们在图和层模型上说明了该框架。我们的结果也可以看作是将已知的图神经网络UAT扩展到一个更一般的设置，该设置也包含层神经网络。此外，我们表明OENN可以进一步扩展到CENN，即范畴等变神经网络，它给出了等变神经网络的一般形式以及等变通用逼近定理，使我们能够利用数据中的范畴对称性（例如，多个对象上的非可逆对称性以及这些对称性上的合成关系）。
英文摘要
Symmetry is everywhere in nature and society. Geometric deep learning builds architectures respecting group symmetries, whereas topological deep learning organizes computation through cells, incidence relations, and local-to-global structure. In this paper, we extend geometric deep learning beyond simple group actions and unify it with topological deep learning. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant vector bundles over face posets (or face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks. We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In the appendix, we show that OENN can be connected, via the action groupoid Grothendieck construction, to CENN (category-equivariant neural network), which gives the categorical general form of equivariant neural networks, allowing us to leverage categorical symmetry in data and extending geometric deep learning from groups of symmetries to categories of transformations.
CommentsAccepted at ICML 2026 as a spotlight paper with oral presentation
