{"ID":2884473,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.07102","arxiv_id":"2508.07102","title":"Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria","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.","short_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-O...","url_abs":"https://arxiv.org/abs/2508.07102","url_pdf":"https://arxiv.org/pdf/2508.07102v1","authors":"[\"Yang Cao\",\"Yubin Chen\",\"Zhao Song\",\"Jiahao Zhang\"]","published":"2025-08-09T21:10:58Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\",\"cs.CV\"]","methods":"[]","has_code":false}