# Flow map matching with stochastic interpolants: A mathematical framework for consistency models

arXiv: https://arxiv.org/abs/2406.07507 · HTML: https://mlsanity.com/paper/2406.07507/

## Key facts

- Cited by 109 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 54 of its 64 references that are in the map.
- First posted on arXiv 11 Jun 2024.

## 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 54)

- [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)
- [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)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Tune-A-Video: One-Shot Tuning of Image Diffusion Models for Text-to-Video Generation](https://mlsanity.com/paper/2212.11565/) (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)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)

## Built on by (most cited first, 30 of 109)

- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [One-Step Diffusion Distillation through Score Implicit Matching](https://mlsanity.com/paper/2410.16794/) (2024)
- [One-step Latent-free Image Generation with Pixel Mean Flows](https://mlsanity.com/paper/2601.22158/) (2026)
- [Categorical Flow Maps](https://mlsanity.com/paper/2602.12233/) (2026)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [Meta Flow Maps enable scalable reward alignment](https://mlsanity.com/paper/2601.14430/) (2026)
- [Much Ado About Noising: Dispelling the Myths of Generative Robotic Control](https://mlsanity.com/paper/2512.01809/) (2025)
- [Noise Hypernetworks: Amortizing Test-Time Compute in Diffusion Models](https://mlsanity.com/paper/2508.09968/) (2025)
- [Transition Models: Rethinking the Generative Learning Objective](https://mlsanity.com/paper/2509.04394/) (2025)
- [pi-Flow: Policy-Based Few-Step Generation via Imitation Distillation](https://mlsanity.com/paper/2510.14974/) (2025)
- [Diamond Maps: Efficient Reward Alignment via Stochastic Flow Maps](https://mlsanity.com/paper/2602.05993/) (2026)
- [Discrete Flow Maps](https://mlsanity.com/paper/2604.09784/) (2026)
- [AnyFlow: Any-Step Video Diffusion Model with On-Policy Flow Map Distillation](https://mlsanity.com/paper/2605.13724/) (2026)
- [Reinforce Adjoint Matching: Scaling RL Post-Training of Diffusion and Flow-Matching Models](https://mlsanity.com/paper/2605.10759/) (2026)
- [Joint Distillation for Fast Likelihood Evaluation and Sampling in Flow-based Models](https://mlsanity.com/paper/2512.02636/) (2025)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [FACM: Flow-Anchored Consistency Models](https://mlsanity.com/paper/2507.03738/) (2025)
- [Ultra-Fast Language Generation via Discrete Diffusion Divergence Instruct](https://mlsanity.com/paper/2509.25035/) (2025)
- [Flow Map Distillation Without Data](https://mlsanity.com/paper/2511.19428/) (2025)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025)
- [Salt: Self-Consistent Distribution Matching with Cache-Aware Training for Fast Video Generation](https://mlsanity.com/paper/2604.03118/) (2026)
- [Ideas in Inference-time Scaling can Benefit Generative Pre-training Algorithms](https://mlsanity.com/paper/2503.07154/) (2025)
- [Posterior Refinement: Fast Language Generation via Any-Order Flow Maps](https://mlsanity.com/paper/2606.24773/) (2026)
- [Parallel Decoding Distillation for Fast Image and Video Generation](https://mlsanity.com/paper/2607.26004/) (2026)
- [DiffRatio: Training One-Step Diffusion Models Without Teacher Supervision](https://mlsanity.com/paper/2502.08005/) (2025)
- [Riemannian MeanFlow](https://mlsanity.com/paper/2602.07744/) (2026)
- [Scaling Categorical Flow Maps](https://mlsanity.com/paper/2605.07820/) (2026)
- [AlignFlow: Improving Flow-based Generative Models with Semi-Discrete Optimal Transport](https://mlsanity.com/paper/2510.15038/) (2025)
- [Skywork UniPic 3.0: Unified Multi-Image Composition via Sequence Modeling](https://mlsanity.com/paper/2601.15664/) (2026)

## Abstract (arXiv)

Generative models based on dynamical equations such as flows and diffusions offer exceptional sample quality, but require computationally expensive numerical integration during inference. The advent of consistency models has enabled efficient one-step or few-step generation, yet despite their practical success, a systematic understanding of their design has been hindered by the lack of a comprehensive theoretical framework. Here we introduce Flow Map Matching (FMM), a principled framework for learning the two-time flow map of an underlying dynamical generative model, thereby providing this missing mathematical foundation. Leveraging stochastic interpolants, we propose training objectives both for distillation from a pre-trained velocity field and for direct training of a flow map over an interpolant or a forward diffusion process. Theoretically, we show that FMM unifies and extends a broad class of existing approaches for fast sampling, including consistency models, consistency trajectory models, and progressive distillation. Experiments on CIFAR-10 and ImageNet-32 highlight that our approach can achieve sample quality comparable to flow matching while reducing generation time by a factor of 10-20.