Flow-Based Global Proposals for Monte Carlo Sampling in SU(2) Lattice Gauge Theory

We propose a formally valid machine-learning-assisted global proposal mechanism for Monte Carlo sampling in lattice gauge theory. The construction is based on a coupling-flow update on the SU(2) lattice-link manifold, in which active links are transformed conditionally on a frozen-link background. For fixed frozen links, the proposal is explicitly invertible and preserves the product Haar measure, so it can be embedded into a Metropolis-Hastings correction without requiring an explicit model of the full proposal density. We implement the method in two-dimensional pure SU(2) lattice gauge theory and benchmark it against a baseline local Metropolis algorithm used as a controlled reference kernel. In the present testbed, the learned proposal reproduces the target ensemble within statistical resolution across the tested configurations. In matched local-step comparisons, the learned proposal reproduces the target ensemble at a quality comparable to the baseline, but does not outperform the pure local baseline in the conservative matched-step case examined with seed-level statistics within this proof-of-principle setup. At the same time, a favorable mixed-step hybrid configuration yields a modest improvement in effective sample size per unit runtime. Because the learned transformation remains in a near-identity regime, the present results should be interpreted as a proof-of-principle demonstration of formal correctness and limited, configuration-dependent efficiency gain within a controlled comparison, rather than as evidence of superiority over optimized conventional update schemes. This work provides a concrete foundation for extending machine-learned nonlocal updates to larger lattices and non-Abelian gauge theories relevant to lattice QCD.