研究解析规范等变图神经网络的闭环几何结构
Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks
一篇理论味很浓的论文：把 GNN 深度到底在做什么讲清楚了，做格点规范场或等变网络的值得看。
arXiv 论文研究了规范等变图神经网络通过局部消息传递生成的闭合格点路径集合，且该分析不依赖网络训练。解析与数值结果显示轮廓长度快速增长，而空间扩张慢得多，更深的网络会在相对紧凑的区域内反复走出行长路径。费米子可观测量的 Loop 展开为这种层级结构提供了物理解释，说明空间局部性与对邻近规范场的高阶依赖可以兼容。论文指出网络深度在扩展空间覆盖与丰富表示两方面扮演互补角色。
Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks
Gauge-equivariant graph neural networks generate ordered products of gauge links through local message passing, but the geometry of these internal structures is not captured by the receptive field alone. We investigate the statistical ensemble of closed lattice walks generated by recursive node- and edge-feature operations, independently of network training. Analytical and numerical results reveal a separation between rapid growth of contour length and much slower spatial expansion: deeper compositions generate increasingly long paths through repeated traversal of relatively compact regions. Loop expansions of fermionic observables provide a physical interpretation of this hierarchy, showing how spatial locality remains compatible with high-order dependence on nearby gauge fields. These findings identify complementary roles of network depth in extending spatial reach and enriching the representations constructed within that reach, providing a geometrical perspective on learning lattice gauge observables.