Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.

翻译：局部规范对称性构成了基本相互作用和强关联量子物质的基础，然而现有的机器学习方法缺乏一种通用的、原理性的框架来学习依赖于局部位点的对称性，特别是针对本质非局域可观测量。本文提出了一种规范等变图神经网络，通过引入矩阵值、规范协变特征和对称性兼容的更新机制，将非阿贝尔对称性直接嵌入消息传递过程，从而将等变学习从全局对称性扩展到完全局域对称性。在该框架中，消息传递实现了穿过格点的规范协变输运，使得非局域关联和环状结构能够从局域操作中自然涌现。我们在纯规范、规范-物质以及动力学体系中验证了该方法，确立了规范等变消息传递作为受局域对称性支配的系统中的通用学习范式。