Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD
This work addresses a critical bottleneck in computational physics for lattice QCD researchers by offering a more efficient sampling method, though it is incremental as it builds on normalizing flows with specific optimizations.
The paper tackles the challenge of sampling Boltzmann distributions in lattice field theories, which suffer from multimodality and long-range correlations, by proposing sparse triangular transport maps that exploit conditional independence structures, achieving linear time complexity in lattice size and outperforming Hybrid Monte Carlo and RealNVP in a controlled 2D φ⁴ setting.
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).