# CAT-Flow: Curvature-Adaptive sTeps for Flow Matching

arXiv: https://arxiv.org/abs/2609.01746 · HTML: https://mlsanity.com/paper/2609.01746/

## 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 39 of its 46 references that are in the map.
- First posted on arXiv 1 Sep 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, 30 of 39)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [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)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space](https://mlsanity.com/paper/2506.15742/) (2025)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [MasaCtrl: Tuning-Free Mutual Self-Attention Control for Consistent Image Synthesis and Editing](https://mlsanity.com/paper/2304.08465/) (2023)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [DiffusionDB: A Large-scale Prompt Gallery Dataset for Text-to-Image Generative Models](https://mlsanity.com/paper/2210.14896/) (2022)
- [SV3D: Novel Multi-view Synthesis and 3D Generation from a Single Image using Latent Video Diffusion](https://mlsanity.com/paper/2403.12008/) (2024)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [HPSv3: Towards Wide-Spectrum Human Preference Score](https://mlsanity.com/paper/2508.03789/) (2025)
- [Stable Flow: Vital Layers for Training-Free Image Editing](https://mlsanity.com/paper/2411.14430/) (2024)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [AdaFlow: Imitation Learning with Variance-Adaptive Flow-Based Policies](https://mlsanity.com/paper/2402.04292/) (2024)
- [TurboEdit: Instant text-based image editing](https://mlsanity.com/paper/2408.08332/) (2024)
- [FS-DFM: Fast and Accurate Long Text Generation with Few-Step Diffusion Language Models](https://mlsanity.com/paper/2509.20624/) (2025)
- [ReFlex: Text-Guided Editing of Real Images in Rectified Flow via Mid-Step Feature Extraction and Attention Adaptation](https://mlsanity.com/paper/2507.01496/) (2025)
- [Image Quality Assessment: From Human to Machine Preference](https://mlsanity.com/paper/2503.10078/) (2025)
- [Bellman Optimal Stepsize Straightening of Flow-Matching Models](https://mlsanity.com/paper/2312.16414/) (2023)
- [Adams Bashforth Moulton Solver for Inversion and Editing in Rectified Flow](https://mlsanity.com/paper/2503.16522/) (2025)
- [CurveFlow: Curvature-Guided Flow Matching for Image Generation](https://mlsanity.com/paper/2508.15093/) (2025)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [SViM3D: Stable Video Material Diffusion for Single Image 3D Generation](https://mlsanity.com/paper/2510.08271/) (2025)
- [Rethinking Image Super-Resolution from Training Data Perspectives](https://mlsanity.com/paper/2409.00768/) (2024)
- [UniFlow-Audio: Unified Flow Matching for Audio Generation from Omni-Modalities](https://mlsanity.com/paper/2509.24391/) (2025)
- [A study of deep perceptual metrics for image quality assessment](https://mlsanity.com/paper/2202.08692/) (2022)

## Abstract (arXiv)

Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.