神经三角输运映射：格点量子色动力学采样的新方法
Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD
AI总结：
针对格点场论采样难题，提出稀疏三角输运映射，利用条件独立结构实现线性复杂度，并在二维φ⁴模型中验证了其优于HMC和RealNVP的性能。
摘要：
格点场论是计算物理学的基本试验场；然而，由于其多模态性和长程相关性，对其玻尔兹曼分布进行采样仍然具有挑战性。虽然归一化流提供了一种有前景的替代方案，但将其应用于大格点时，常常受到内存需求过高和保持足够模型表达能力难度的限制。我们提出稀疏三角输运映射，该方法利用单调修正神经网络（MRNN），显式地利用周期边界条件下格点图的条件独立结构。我们引入了一个三角输运映射的综合框架，该框架在精确稀疏性（尊重目标分布中的边际条件独立性）和近似稀疏性（无填充的计算可行性）之间权衡。将每个三角映射分量限制为局部过去，可实现逐点并行评估和格点规模$N$的线性时间复杂度，同时保持表达性、可逆的结构。以二维$\phi^4$作为受控设置，我们分析了节点标记（排序）如何影响三角映射的稀疏性和性能。我们将所提方法与混合蒙特卡洛（HMC）和已建立的流方法（RealNVP）进行了比较。
英文摘要：
Lattice field theories are fundamental testbeds for computational physics; yet, sampling their Boltzmann distributions remains challenging due to multimodality and long-range correlations. While normalizing flows offer a promising alternative, their application to large lattices is often constrained by prohibitive memory requirements and the challenge of maintaining sufficient model expressivity. We propose sparse triangular transport maps that explicitly exploit the conditional independence structure of the lattice graph under periodic boundary conditions using monotone rectified neural networks (MRNN). We introduce a comprehensive framework for triangular transport maps that navigates the fundamental trade-off between \emph{exact sparsity} (respecting marginal conditional independence in the target distribution) and \emph{approximate sparsity} (computational tractability without fill-ins). Restricting each triangular map component to a local past enables site-wise parallel evaluation and linear time complexity in lattice size $N$, while preserving the expressive, invertible structure. Using $ϕ^4$ in two dimensions as a controlled setting, we analyze how node labelings (orderings) affect the sparsity and performance of triangular maps. We compare against Hybrid Monte Carlo (HMC) and established flow approaches (RealNVP).
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