# Patched Flow Matching: Generative Wall-Pressure Reconstruction Beyond Training-Domain Scales from Sparse Sensors

arXiv: https://arxiv.org/abs/2606.22084 · HTML: https://mlsanity.com/paper/2606.22084/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 16 of its 45 references that are in the map.
- First posted on arXiv 20 Jun 2026.

## 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, 16 of 16)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation](https://mlsanity.com/paper/2311.07896/) (2023)
- [On the Edge of Memorization in Diffusion Models](https://mlsanity.com/paper/2508.17689/) (2025)
- [Learning Image Priors through Patch-based Diffusion Models for Solving Inverse Problems](https://mlsanity.com/paper/2406.02462/) (2024)
- [CoNFiLD-inlet: Synthetic Turbulence Inflow Using Generative Latent Diffusion Models with Neural Fields](https://mlsanity.com/paper/2411.14378/) (2024)
- [Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments](https://mlsanity.com/paper/2506.00214/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [Diagnosing and Improving Diffusion Models by Estimating the Optimal Loss Value](https://mlsanity.com/paper/2506.13763/) (2025)
- [Predicting the near-wall region of turbulence through convolutional neural networks](https://mlsanity.com/paper/2107.07340/) (2021)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)

## Abstract (arXiv)

Characterizing the complete wall-pressure spectrum in turbulent wall-bounded flows requires simultaneous access to the viscous-scale high-wavenumber content and the outer-layer low-wavenumber content -- a requirement that neither short-domain direct numerical simulation (DNS) nor sparse experimental measurements alone can satisfy. We propose Patched Flow Matching (Patched FM), a generative framework that fuses these two complementary sources by learning a patch-local prior over inner-scaled wall-pressure statistics from short-domain DNS and assimilating sparse sensor measurements at inference time through training-free posterior sampling. The patch-additive decomposition of the flow matching vector field decouples the generative prior from the global domain size, enabling reconstruction on domains arbitrarily larger than the training configuration. By expressing the patch prior in inner-scaled coordinates, where high-wavenumber wall-pressure statistics are approximately Reynolds-number invariant, the framework extends to higher Reynolds numbers through hierarchical transfer learning with as few as 500 short-domain snapshots (2.5% of the base training data) at a fraction of the scratch-training cost. Applied to compressible channel-flow DNS at Reτ = 180, 500, and 1000, Patched FM reconstructs full-resolution wall-pressure fields on a domain four times larger than the training configuration (LxL = 16πδ versus LxS = 4πδ) from sensor coverage as low as 0.25%, recovering the low-wavenumber spectral content inaccessible to short-domain DNS with high fidelity in both streamwise and spanwise directions. Zero-shot generalization to unseen Reynolds numbers and ablation studies further confirm the role of inner scaling as a physical prerequisite for data-efficient Reynolds-number transfer.