提出高阶平均流模型，实现高效一步采样与强表达能力。
Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria
- 引入平均加速度场扩展一阶平均流，提升建模动态性。
- 证明可由TC⁰类电路实现，具备强计算表达能力。
- 提出近似注意力算法，支持大规模高效训练与采样。
生成建模近年得益于无模拟范式如流匹配，特别是平均流（MeanFlow）框架，通过用平均速度替代瞬时速度，实现高效的单步采样。本文首次对二阶平均流（Second-Order MeanFlow）进行理论研究，该模型将平均加速度场引入目标函数。我们首先证明平均加速度满足广义一致性条件，支持稳定的一步采样与可计算的损失函数。接着通过电路复杂度分析刻画其表达能力：在弱假设下，二阶平均流采样过程可由统一阈值电路在$\mathsf{TC}^0$类中实现。最后，基于快速近似注意力计算，推导出可证明高效的实现准则：在时间$ n^{2+o(1)} $内，可将注意力操作近似到$1/\mathrm{poly}(n)$误差。这些结果为兼具丰富动力学与实际采样效率的高阶流匹配模型奠定了理论基础。
原文摘要 · Abstract (English)
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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