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

Abstract

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.

31 pages, 22 figures

References in corpus (17)

• Exchange Monte Carlo Method and Application to Spin Glass Simulations
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Computational Studies of Quantum Spin Systems
• Full nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks
• Critical slowing down and error analysis in lattice QCD simulations
• Flow-based generative models for Markov chain Monte Carlo in lattice field theory
• Equivariant flow-based sampling for lattice gauge theory
• Sampling using $SU(N)$ gauge equivariant flows
• Probability-Changing Cluster Algorithm for Two-Dimensional XY and Clock Models
• Finite-size scaling method for the Berezinskii-Kosterlitz-Thouless transition
• Simulating spin models on GPU
• Flow-based sampling in the lattice Schwinger model at criticality
• Learning Lattice Quantum Field Theories with Equivariant Continuous Flows
• Diffusion Models as Stochastic Quantization in Lattice Field Theory
• Unification of Symmetries Inside Neural Networks: Transformer, Feedforward and Neural ODE
• Stochastic quantization and diffusion models
• Lecture Notes on Normalizing Flows for Lattice Quantum Field Theories