# From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity

arXiv: https://arxiv.org/abs/2512.02826 · HTML: https://mlsanity.com/paper/2512.02826/

## Key facts

- Cited by 8 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 39 of its 47 references that are in the map.
- First posted on arXiv 2 Dec 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 39)

- [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)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Wan: Open and Advanced Large-Scale Video Generative Models](https://mlsanity.com/paper/2503.20314/) (2025)
- [Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasets](https://mlsanity.com/paper/2311.15127/) (2023)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space](https://mlsanity.com/paper/2506.15742/) (2025)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Reconstruction vs. Generation: Taming Optimization Dilemma in Latent Diffusion Models](https://mlsanity.com/paper/2501.01423/) (2025)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models](https://mlsanity.com/paper/2404.07724/) (2024)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Open-Sora 2.0: Training a Commercial-Level Video Generation Model in 200k](https://mlsanity.com/paper/2503.09642/) (2025)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)

## Built on by (most cited first, 8 of 8)

- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [CF-VLA: Efficient Coarse-to-Fine Action Generation for Vision-Language-Action Policies](https://mlsanity.com/paper/2604.24622/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Support-Conditioned Flow Matching Is Kernel Smoothing](https://mlsanity.com/paper/2605.13386/) (2026)
- [Breaking the Lock-in: Diversifying Text-to-Image Generation via Representation Modulation](https://mlsanity.com/paper/2606.06813/) (2026)
- [A Theoretical Analysis of Memory and Overfitting Phenomena in Stochastic Interpolation Models](https://mlsanity.com/paper/2606.08554/) (2026)
- [CAT-Flow: Curvature-Adaptive sTeps for Flow Matching](https://mlsanity.com/paper/2609.01746/) (2026)
- [Two Halves are More than One: Phase-wise Velocity Distillation for Fast and High-Quality Image Generation](https://mlsanity.com/paper/2610.08070/) (2026)

## Abstract (arXiv)

Flow-based diffusion models have emerged as a leading paradigm for training generative models across images and videos. However, their memorization-generalization behavior remains poorly understood. In this work, we revisit the flow matching (FM) objective and study its marginal velocity field, which admits a closed-form expression, allowing exact computation of the oracle FM target. Analyzing this oracle velocity field reveals that flow-based diffusion models inherently formulate a two-stage training target: an early stage guided by a mixture of data modes, and a later stage dominated by the nearest data sample. The two-stage objective leads to distinct learning behaviors: the early navigation stage generalizes across data modes to form global layouts, whereas the later refinement stage increasingly memorizes fine-grained details. Leveraging these insights, we explain the effectiveness of practical techniques such as timestep-shifted schedules, classifier-free guidance intervals, and latent space design choices. Our study deepens the understanding of diffusion model training dynamics and offers principles for guiding future architectural and algorithmic improvements. Our project page is available at: https://maps-research.github.io/from-navigation-to-refinement/.