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

arXiv: https://arxiv.org/abs/2508.15318 · HTML: https://mlsanity.com/paper/2508.15318/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 21 Aug 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 18 of 18)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Extending Context Window of Large Language Models via Positional Interpolation](https://mlsanity.com/paper/2306.15595/) (2023)
- [Do Vision Transformers See Like Convolutional Neural Networks?](https://mlsanity.com/paper/2108.08810/) (2021)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [cuDNN: Efficient Primitives for Deep Learning](https://mlsanity.com/paper/1410.0759/) (2014)
- [Efficient and Unbiased Sampling from Boltzmann Distributions via Variance-Tuned Diffusion Models](https://mlsanity.com/paper/2505.21005/) (2025)

## Built on by (most cited first, 2 of 2)

- [Capturing the Topological Phase Transition and Thermodynamics of the 2D XY Model via Manifold-Aware Score-Based Generative Modeling](https://mlsanity.com/paper/2602.07548/) (2026)
- [Diffusion-warm sampling of the XY model enables fast thermalization at scale](https://mlsanity.com/paper/2606.30773/) (2026)

## Abstract (arXiv)

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 L2 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.