用规范不变图神经网络模拟规范场理论，高效捕捉非局域关联。
Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories
- 以威尔逊环为输入，显式保证规范对称性，避免冗余自由度。
- 在Z2和U(1)模型上准确预测全局与局域物理量，含强耦合非局域效应。
- 可作为半经典动力学高效代理，免去重复对角化，适合大规模演化模拟。
局部规范结构在凝聚态系统与量子平台中起核心作用，常作为强关联相和工程动力学的有效描述。本文提出一种规范不变的图神经网络（GNN）架构，通过局部规范不变输入（如威尔逊环）显式施加对称性，并在整个消息传递过程中保持规范对称性，从而消除冗余规范自由度，同时保留表达能力。我们在$\mathbb{Z}_2$和$\mathrm{U}(1)$格点规范模型上进行基准测试，即使在规范-物质耦合引发的非局域相关下，仍能准确预测全局可观测量与空间分辨量。进一步证明，所学习模型可作为$\mathrm{U}(1)$量子链接模型中半经典动力学的高效代理，实现稳定可扩展的时间演化，无需重复费米子对角化，且忠实地再现局部动力学与统计关联。这些结果确立了规范不变消息传递作为学习与模拟阿贝尔格点规范系统的紧凑且物理根基稳固的框架。
原文摘要 · Abstract (English)
Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both $\mathbb{Z}_2$ and $\mathrm{U}(1)$ lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in $\mathrm{U}(1)$ quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.
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