Efficient Identification of Critical Transitions via Flow Matching: A Scalable Generative Approach for Many-Body Systems

We propose a machine learning framework based on Flow Matching (FM) to identify critical properties in many-body systems efficiently. Using the 2D XY model as a benchmark, we demonstrate that a single network, trained only on configurations from a small ($32\times 32$) lattice at sparse temperature points, effectively generalizes across both temperature and system size. This dual generalization enables two primary applications for large-scale computational physics: (i) a rapid "train-small, predict-large" strategy to locate phase transition points for significantly larger systems ($128\times 128$) without retraining, facilitating efficient finite-size scaling analysis; and (ii) the fast generation of high-fidelity, decorrelated initial spin configurations for large-scale Monte Carlo simulations, providing a robust starting point that bypasses the long thermalization times of traditional samplers. These capabilities arise from the combination of the Flow Matching framework, which learns stable probability-flow vector fields, and the inductive biases of the U-Net architecture that capture scale-invariant local correlations. Our approach offers a scalable and efficient tool for exploring the thermodynamic limit, serving as both a rapid explorer for phase boundaries and a high-performance initializer for high-precision studies.