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

TL;DR

This work introduces a Flow Matching framework combined with a U‑Net to efficiently identify critical transitions in many‑body systems. Trained on a small $32\times32$ lattice, the model generalizes across temperature and extrapolates to much larger lattices, enabling a train‑small, predict‑large workflow and fast decorrelated initial configurations for large‑scale Monte Carlo simulations. Using the 2D XY model, the approach demonstrates accurate interpolation across temperatures, robust but qualitative finite‑size scaling behavior, and substantial computational savings by avoiding retraining for each system size. The results suggest a scalable path to thermodynamic‑limit studies and hybrid workflows where FM seeding reduces thermalization times, with potential extensions to lattice field theories and quantum problems.

Abstract

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.

Figures (20)

• Figure 1: Schematic diagram of our work. Using Flow Matching with U-Net ronneberger2015unet inductive biases, we learn a deterministic probability-flow sampler that—trained only on $L_0\times L_0$ lattices—simultaneously interpolates across temperatures and extrapolates (dashed lines) to physically consistent configurations of $L_1\times L_1$ lattices with $L_1\gg L_0$, revealing scale-invariant rules and enabling a fast and stable exploration of critical phenomena near the critical temperature, $T_c$. Here $\chi$ denotes the susceptibility, and the green shadow represents the critical regime.
• Figure 2: A naive Flow Matching example. When Flow Matching is trained as a deterministic ODE, trajectories from $\mathbf{v_{\theta}}$ deviate from the straight interpolation in Eq. \ref{['eq_interpolation_path']}. This is because $\mathbf{ v_\theta}$ learns the expectation of the conditional velocity, $\mathbf{v_t}(\mathbf{x}_t|\phi)$, induced by the random pairings of $(\mathbf{x}_0,\mathbf{x}_1)$ during training (left panel). Consequently, the first ODE iteration step (middle panel) points toward the target mean; with a finite number of iterations, inference preferentially produces samples near this mean, thereby underrepresenting the target variance and fluctuations.
• Figure 3: Thermodynamic observables for the $32\times32$ XY model, demonstrating the Flow Matching (FM) model's ability to increase sampling density via interpolation. The panels show (a) energy per site $E(T)$ and magnetization $m(T)$, and (b) spin stiffness $\rho_s(T)$ and vortex density $\rho_v(T)$. The sparse points represent MCMC (MC) data, which serves as the training set, sampled at a coarse temperature interval of $\Delta T = 0.10$. Each of these points was generated from an ensemble of $5000$ configurations. The denser set of points shows the results from FM, evaluated on a tenfold denser grid ($\Delta T = 0.01$), confirming agreement with and smooth interpolation between the training points.
• Figure 4: Susceptibility $\chi(T)$ for a $32\times32$ XY lattice. The points show results from MCMC (MC) simulations, which serve as both the ground truth and the training data for our Flow Matching (FM) model. The three panels demonstrate the model's performance when trained on this MCMC data at varying levels of precision: (a) a dense grid with temperature spacing $\Delta T = 0.05$, (b) $\Delta T = 0.10$, and (c) a sparse grid with $\Delta T = 0.50$. The training data consists of $5000$ configurations per temperature point. In all cases, the FM model is evaluated on a much finer grid ($\Delta T = 0.01$), showing it successfully reproduces the BKT-related features even when trained on the sparsest data set (In (c), Monte Carlo training data is available only at the four temperatures outside the phase transition point.). Since our method directly generates spin configurations rather than predicting physical quantities, it is well-suited for generating high-quality initial spin configurations for Monte Carlo simulations that require high accuracy.
• Figure 5: Size extrapolation results for the Flow Matching (FM) model. The model was trained on data from a $32\times32$ system ($T\in[0.1,2.0]$, $\Delta T=0.01$, 1500 configs/T) and then used to generate configurations for a $64\times64$ lattice. The generated observables—(a) magnetization, (b) susceptibility, (c) spin stiffness, and (d) vortex density—are shown to be in good agreement with reference MCMC results for the larger system.