Efficient identification of critical regions via Flow Matching-based Monte Carlo initialization

Markov chain Monte Carlo (MCMC) is a standard tool for studying many-body systems, but its practical cost can become substantial, especially when simulations must be repeated across temperatures and lattice sizes or near transition regions where equilibration becomes increasingly difficult. In this work, we introduce a Flow Matching (FM) framework. It is not a standalone replacement for equilibrium Monte Carlo. Instead, we present it as a scalable, physically informed initializer for downstream MCMC simulations. We use a U-Net architecture. The FM model is trained on small-system configurations of the 2D XY model and then deployed across unseen temperatures and larger lattice sizes. FM-generated configurations preserve the correct qualitative physical trends across temperature and system size. This makes them suitable warm-start states for subsequent Monte Carlo refinement. Observables computed from FM-generated samples primarily serve as diagnostics of initializer quality, not as precision equilibrium estimates. The regression-based L_2 objective suppresses variance and limits the accuracy of fluctuation-sensitive observables. Examples include susceptibility and spin stiffness. Still, the model captures sufficient local statistical structure to yield physically aligned initial states across a broad range of conditions. These results support a reusable hybrid FM--MCMC workflow. The one-time FM training cost can be amortized across temperatures and lattice sizes. The generated warm-start configurations then reduce the burden of initializing large-scale Monte Carlo simulations. Our findings show that Flow Matching can support efficient exploration of transition regions in many-body systems by providing reusable warm-start configurations for downstream Monte Carlo simulations.