# Carr'e du champ flow matching: better quality-generalisation tradeoff in generative models

arXiv: https://arxiv.org/abs/2510.05930 · HTML: https://mlsanity.com/paper/2510.05930/

## Key facts

- Cited by 14 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 30 of its 50 references that are in the map.
- First posted on arXiv 7 Oct 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, 30 of 30)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [Motion Flow Matching for Human Motion Synthesis and Editing](https://mlsanity.com/paper/2312.08895/) (2023)
- [Memorization to Generalization: Emergence of Diffusion Models from Associative Memory](https://mlsanity.com/paper/2505.21777/) (2025)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [Losing dimensions: Geometric memorization in generative diffusion](https://mlsanity.com/paper/2410.08727/) (2024)
- [Interpretable statistical representations of neural population dynamics and geometry](https://mlsanity.com/paper/2304.03376/) (2023)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)

## Built on by (most cited first, 14 of 14)

- [Gradient Variance Reveals Failure Modes in Flow-Based Generative Models](https://mlsanity.com/paper/2510.18118/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Optimal Transport Flow Matching by Design](https://mlsanity.com/paper/2606.04092/) (2026)
- [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026)
- [Difficulty-Calibrated Interpolation Paths for Conditional Flow Matching](https://mlsanity.com/paper/2608.21286/) (2026)
- [Shortest-Path Flow Matching with Mixture-Conditioned Bases for OOD Generalization to Unseen Conditions](https://mlsanity.com/paper/2601.11827/) (2026)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Diffusion Maps is not Dimensionality Reduction](https://mlsanity.com/paper/2603.28037/) (2026)
- [Neural Point-Forms](https://mlsanity.com/paper/2605.15524/) (2026)
- [Why Are DMD Students Lazy? Understanding the Copying Behavior in Few-Step Distillation](https://mlsanity.com/paper/2606.02237/) (2026)
- [Points as Tori: Fast Pointwise Signed Distance for Point Clouds](https://mlsanity.com/paper/2607.16946/) (2026)
- [DriftSE: Speech Enhancement with Generative Drifting](https://mlsanity.com/paper/2609.12252/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)
- [Intrinsic Geometry Matching: learning observation-equivariant process geometry](https://mlsanity.com/paper/2610.16995/) (2026)

## Abstract (arXiv)

Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carr'e du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.