发表机构
Berlin Institute for the Foundations of Learning and Data (BIFOLD); Technische Universität Berlin; Bergische Universität Wuppertal; Deutsches Elektronen-Synchrotron (DESY); Computation-Based Science and Technology Research Center, The Cyprus Institute; Department of Electrical Engineering (ESAT), KU Leuven; RIKEN Center for Advanced Intelligence Project (AIP)（柏林学习与数据基础研究所; 柏林工业大学; 伍珀塔尔大学; 德国电子同步辐射研究中心; 塞浦路斯研究院计算科学与技术研究中心; 荷语鲁汶大学电气工程系; 理化学研究所先进智能项目中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对格点规范理论，提出多层环面空间采样器，通过粗到细分解精确满足比安奇约束，在弱耦合区域显著优于链接空间基线。
AI 中文摘要
格点规范理论是一类重要的物理模型，具有高维结构化分布，支撑着粒子物理、核物理和凝聚态物理中的第一性原理计算。生成建模的最新进展为格点规范理论中玻尔兹曼分布的采样开辟了新途径，有望缓解传统蒙特卡罗方法的局限性，包括临界慢化和拓扑冻结。然而，格点规范理论的生成采样器通常在链接空间中构建，在弱耦合区域，相关性变得越来越长程，这给学习带来了挑战。环面提供了更自然的表示，因为作用量在这些变量中是局部的。然而，精确的比安奇约束将它们限制在较低维的流形上，使直接生成建模变得复杂。我们引入了一种多层归一化流构造，直接在环面空间中采样，同时精确满足这些约束。关键思想是一种从粗到细的分解，将全局耦合的约束问题转化为一系列局部求解：在每次细化时，每个确定的环面最多依赖于四个新生成的变量，而约束问题的大小与格点尺寸无关。我们在二维和四维的$U(1)$以及二维的$SU(2)$上验证了该构造。我们的多层环面空间采样器（PSS）显著优于链接空间基线，且优势在弱耦合区域（规范耦合变小时）增大。这一区域对生成采样尤其具有挑战性，在渐近自由的规范理论中，与连续统研究相关。
英文摘要
Lattice gauge theories are an important class of physical models with high-dimensional structured distributions that underpin first-principles calculations in particle, nuclear, and condensed-matter physics. Recent advances in generative modeling have opened new avenues for sampling Boltzmann distributions in lattice gauge theories, offering the potential to alleviate limitations of traditional Monte Carlo methods, including critical slowing down and topological freezing. However, generative samplers for lattice gauge theories are typically constructed in link space, where correlations become increasingly long-ranged toward weak coupling, posing a challenge for learning. Plaquettes provide a more natural representation, as the action is local in these variables. However, exact Bianchi constraints restrict them to a lower-dimensional manifold, complicating direct generative modeling. We introduce a multilevel normalizing-flow construction that samples directly in plaquette space while satisfying these constraints exactly. The key idea is a coarse-to fine factorization that transforms a globally coupled constraint problem into a sequence of local solves: at each refinement, every determined plaquette depends on at most four newly generated variables, while the size of the constraint problem remains independent of the lattice size. We validate the construction for $U(1)$ in two and four dimensions and for $SU(2)$ in two dimensions. Our multilevel Plaquette-Space Sampler (PSS) substantially outperforms link-space baselines, with the advantage increasing toward weak coupling, where the gauge coupling becomes small. This regime is particularly challenging for generative sampling and, in asymptotically free gauge theories, is relevant to continuum studies.
Comments9 pages main text, 38 pages total, 10 figures, 9 tables
