# Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria

arXiv: https://arxiv.org/abs/2508.07102 · HTML: https://mlsanity.com/paper/2508.07102/

## Key facts

- Cited by 5 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 62 of its 66 references that are in the map.
- First posted on arXiv 9 Aug 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 62)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [Looped Transformers are Better at Learning Learning Algorithms](https://mlsanity.com/paper/2311.12424/) (2023)
- [Consistency Models Made Easy](https://mlsanity.com/paper/2406.14548/) (2024)
- [Saturated Transformers are Constant-Depth Threshold Circuits](https://mlsanity.com/paper/2106.16213/) (2021)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Fast Attention Requires Bounded Entries](https://mlsanity.com/paper/2302.13214/) (2023)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)

## Built on by (most cited first, 5 of 5)

- [Your Vision-Language Model Can't Even Count to 20: Exposing the Failures of VLMs in Compositional Counting](https://mlsanity.com/paper/2510.04401/) (2025)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)
- [Too Easily Fooled? Prompt Injection Breaks LLMs on Frustratingly Simple Multiple-Choice Questions](https://mlsanity.com/paper/2508.13214/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [Tensor Gauge Flow Models](https://mlsanity.com/paper/2511.17616/) (2025)

## Abstract (arXiv)

Generative modelling has seen significant advances through simulation-free paradigms such as Flow Matching, and in particular, the MeanFlow framework, which replaces instantaneous velocity fields with average velocities to enable efficient single-step sampling. In this work, we introduce a theoretical study on Second-Order MeanFlow, a novel extension that incorporates average acceleration fields into the MeanFlow objective. We first establish the feasibility of our approach by proving that the average acceleration satisfies a generalized consistency condition analogous to first-order MeanFlow, thereby supporting stable, one-step sampling and tractable loss functions. We then characterize its expressivity via circuit complexity analysis, showing that under mild assumptions, the Second-Order MeanFlow sampling process can be implemented by uniform threshold circuits within the mathsfTC0 class. Finally, we derive provably efficient criteria for scalable implementation by leveraging fast approximate attention computations: we prove that attention operations within the Second-Order MeanFlow architecture can be approximated to within 1/poly(n) error in time n2+o(1). Together, these results lay the theoretical foundation for high-order flow matching models that combine rich dynamics with practical sampling efficiency.