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

• URL: http://arxiv.org/abs/2508.07102v1
• Date: Sat, 09 Aug 2025 21:10:58 GMT
• Title: Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria
• Authors: Yang Cao, Yubin Chen, Zhao Song, Jiahao Zhang,
• Abstract summary: We introduce a theoretical study on Second-Order MeanFlow, a novel extension that incorporates average acceleration fields into the MeanFlow objective.<n>We first establish the feasibility of our approach by proving that the average acceleration satisfies a generalized consistency condition analogous to first-order MeanFlow.<n>We then characterize its expressivity via circuit complexity analysis, showing that the Second-Order MeanFlow sampling process can be implemented by uniform threshold circuits.
• Score: 14.37317011636007
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: 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 $\mathsf{TC}^0$ 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 $1/\mathrm{poly}(n)$ error in time $n^{2+o(1)}$. Together, these results lay the theoretical foundation for high-order flow matching models that combine rich dynamics with practical sampling efficiency.

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