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 $\phi^4$ 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).

翻译：格点场论是计算物理学的基础试验平台；然而，由于其多模态性和长程相关性，对其玻尔兹曼分布的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，但其在大规模格点上的应用常受限于过高的内存需求以及保持足够模型表达能力的挑战。我们提出稀疏三角传输映射，该方法利用单调整流神经网络（MRNN）显式地利用了周期边界条件下格点图的条件独立结构。我们引入了一个全面的三角传输映射框架，该框架在**精确稀疏性**（尊重目标分布中的边缘条件独立性）与**近似稀疏性**（无需填充即可实现计算可行性）之间进行权衡。将每个三角映射分量限制在局部历史范围内，可实现格点位置上的并行评估以及关于格点尺寸 $N$ 的线性时间复杂度，同时保持其富有表达力且可逆的结构。以二维 $\phi^4$ 理论作为受控环境，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。我们将其与混合蒙特卡洛方法（HMC）以及成熟的流方法（RealNVP）进行了比较。