Flow Matching at Scale: A Machine Learning Framework for Efficient Large-Size Sampling of Many-Body Systems

We propose a machine learning framework based on Flow Matching to overcome the scaling limitations of Markov Chain Monte Carlo (MCMC) methods. We demonstrate its capability in the 2D XY model, where a single network, trained only on configurations from a small ($32\times 32$) lattice at sparse temperature points, generates reliable samples for a significantly larger system ($128\times 128$) across a continuous temperature range without retraining. The generated configurations show strong agreement with key thermodynamic observables and correctly capture the signatures of the Berezinskii-Kosterlitz-Thouless (BKT) transition. This dual generalization is enabled by the Flow Matching framework, which allows us to learn a continuous, temperature-conditioned mapping. At the same time, the inductive biases of the underlying CNN architecture ensure that the learned local physical rules are scale-invariant. This "train-small, generate-large" capability offers a powerful and efficient alternative for studying critical phenomena. The method can be directly applied to other classical or quantum many-body systems described by continuous fields on a lattice. Furthermore, this framework can serve as a powerful proposal generator in a hybrid scheme with MCMC, dramatically accelerating high-precision studies of the thermodynamic limit.