Neural Triangular Transport Maps for Sampling in Lattice QCD

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 scalability to large lattices remains a challenge. We propose sparse triangular transport maps that explicitly encode 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). Unlike dense normalizing flows that suffer from $\mathcal{O}(N^2)$ dependencies, our approach leverages locality to reduce complexity to $\mathcal{O}(N)$ while maintaining expressivity. Using $\phi^4$ in two dimensions as a controlled setting, we analyze how node labelings (orderings) affect sparsity and performance of triangular maps. We compare against Hybrid Monte Carlo (HMC) and established flow approaches (RealNVP). Our results suggest that structure-exploiting triangular transports deliver tractable scaling and competitive decorrelation compared to dense or coupling-based flows, while preserving physical symmetries via localized stencils.