# SplitMeanFlow: Interval Splitting Consistency in Few-Step Generative Modeling

arXiv: https://arxiv.org/abs/2507.16884 · HTML: https://mlsanity.com/paper/2507.16884/

## Key facts

- Cited by 26 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 33 of its 33 references that are in the map.
- First posted on arXiv 22 Jul 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 33)

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Robust Speech Recognition via Large-Scale Weak Supervision](https://mlsanity.com/paper/2212.04356/) (2022)
- [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)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [HunyuanVideo: A Systematic Framework For Large Video Generative Models](https://mlsanity.com/paper/2412.03603/) (2024)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Lumiere: A Space-Time Diffusion Model for Video Generation](https://mlsanity.com/paper/2401.12945/) (2024)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Seed-TTS: A Family of High-Quality Versatile Speech Generation Models](https://mlsanity.com/paper/2406.02430/) (2024)
- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [FunASR: A Fundamental End-to-End Speech Recognition Toolkit](https://mlsanity.com/paper/2305.11013/) (2023)
- [Large-scale Self-Supervised Speech Representation Learning for Automatic Speaker Verification](https://mlsanity.com/paper/2110.05777/) (2021)
- [Multistep Consistency Models](https://mlsanity.com/paper/2403.06807/) (2024)
- [AudioX: A Unified Framework for Anything-to-Audio Generation](https://mlsanity.com/paper/2503.10522/) (2025)
- [LAFMA: A Latent Flow Matching Model for Text-to-Audio Generation](https://mlsanity.com/paper/2406.08203/) (2024)
- [AudioLCM: Text-to-Audio Generation with Latent Consistency Models](https://mlsanity.com/paper/2406.00356/) (2024)

## Built on by (most cited first, 26 of 26)

- [AlphaFlow: Understanding and Improving MeanFlow Models](https://mlsanity.com/paper/2510.20771/) (2025)
- [AnyFlow: Any-Step Video Diffusion Model with On-Policy Flow Map Distillation](https://mlsanity.com/paper/2605.13724/) (2026)
- [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025)
- [Parallel Decoding Distillation for Fast Image and Video Generation](https://mlsanity.com/paper/2607.26004/) (2026)
- [Riemannian MeanFlow](https://mlsanity.com/paper/2602.07744/) (2026)
- [Drift-Based Policy Optimization: Native One-Step Policy Learning for Online Robot Control](https://mlsanity.com/paper/2604.03540/) (2026)
- [Exposure Bias Can Alleviate Itself via Directional and Frequency Rectification in Flow Matching](https://mlsanity.com/paper/2606.28226/) (2026)
- [Flow-Map GRPO: Reinforcement Learning for Few-Step Flow-Map Generators via Anchored Stochastic Composition](https://mlsanity.com/paper/2607.00535/) (2026)
- [Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning](https://mlsanity.com/paper/2609.15903/) (2026)
- [Trajectory Consistency for One-Step Generation on Euler Mean Flows](https://mlsanity.com/paper/2602.02571/) (2026)
- [Extending One-Step Image Generation from Class Labels to Text via Discriminative Text Representation](https://mlsanity.com/paper/2604.18168/) (2026)
- [CoFlow: Coordinated Few-Step Flow for Offline Multi-Agent Decision Making](https://mlsanity.com/paper/2605.01457/) (2026)
- [WFR-MFM: One-Step Inference for Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.20606/) (2026)
- [DSFlow: Dual Supervision and Step-Aware Architecture for One-Step Flow Matching Speech Synthesis](https://mlsanity.com/paper/2602.09041/) (2026)
- [Noise-Started One-Step Real-World Super-Resolution via LR-Conditioned SplitMeanFlow and GAN Refinement](https://mlsanity.com/paper/2605.09328/) (2026)
- [CaloTrilogy: Toward a Breakthrough in One-Step, End-to-End, Physics-Guided Shower Generation for Modern Calorimeters](https://mlsanity.com/paper/2606.04165/) (2026)
- [High-Fidelity One-Step Generative Visuomotor Policy via Recursive Correction, Frequency Consistency, and Contrastive Flow Matching](https://mlsanity.com/paper/2607.03865/) (2026)
- [Expanding Flow Maps](https://mlsanity.com/paper/2607.21585/) (2026)
- [ReflexFlow: Rethinking Learning Objective for Exposure Bias Alleviation in Flow Matching](https://mlsanity.com/paper/2512.04904/) (2025)
- [Dual-End Consistency Model](https://mlsanity.com/paper/2602.10764/) (2026)
- [Why Are DMD Students Lazy? Understanding the Copying Behavior in Few-Step Distillation](https://mlsanity.com/paper/2606.02237/) (2026)
- [Distill Locally, Schedule Globally: Flow Maps for Few-Step Text-to-Speech](https://mlsanity.com/paper/2609.36324/) (2026)
- [Kinematic MeanFlow: One-Step Action Generation Policy for Robotic Foundation Models](https://mlsanity.com/paper/2610.00864/) (2026)
- [PickMoment: Continuous-Time Single-Image-to-Video via Learning Deblurring and Blur-to-Video](https://mlsanity.com/paper/2610.01279/) (2026)
- [Parametric Trajectory Distillation for Few-Step Video Generation](https://mlsanity.com/paper/2610.11498/) (2026)
- [Just Weather Scoring: Efficient End-to-end Nowcasting with Distributional Diffusion](https://mlsanity.com/paper/2610.12189/) (2026)

## Abstract (arXiv)

Generative models like Flow Matching have achieved state-of-the-art performance but are often hindered by a computationally expensive iterative sampling process. To address this, recent work has focused on few-step or one-step generation by learning the average velocity field, which directly maps noise to data. MeanFlow, a leading method in this area, learns this field by enforcing a differential identity that connects the average and instantaneous velocities. In this work, we argue that this differential formulation is a limiting special case of a more fundamental principle. We return to the first principles of average velocity and leverage the additivity property of definite integrals. This leads us to derive a novel, purely algebraic identity we term Interval Splitting Consistency. This identity establishes a self-referential relationship for the average velocity field across different time intervals without resorting to any differential operators. Based on this principle, we introduce SplitMeanFlow, a new training framework that enforces this algebraic consistency directly as a learning objective. We formally prove that the differential identity at the core of MeanFlow is recovered by taking the limit of our algebraic consistency as the interval split becomes infinitesimal. This establishes SplitMeanFlow as a direct and more general foundation for learning average velocity fields. From a practical standpoint, our algebraic approach is significantly more efficient, as it eliminates the need for JVP computations, resulting in simpler implementation, more stable training, and broader hardware compatibility. One-step and two-step SplitMeanFlow models have been successfully deployed in large-scale speech synthesis products (such as Doubao), achieving speedups of 20x.