提出一种新生成框架，实现高效稳定的一步生成。
Trajectory Consistency for One-Step Generation on Euler Mean Flows
- 用线性近似替代难优化的轨迹一致性约束
- 图像等任务中采样效率提升50%，训练更快更省内存
- 无需计算雅可比矩阵，适合资源受限场景
我们提出基于流的生成框架——欧拉均值流（Euler Mean Flows, EMF），支持一步和少量步骤生成，并在极低采样成本下实现长程轨迹一致性。核心思想是将难以监督与优化的长期轨迹一致性约束，替换为一个基于流模型半群形式推导出的合理线性近似，从而实现对长时程流映射组合的直接数据监督。在弱正则性假设下，该近似能忠实逼近原目标，同时显著降低优化难度。该方法构建了一个统一的、无需计算雅可比向量积（JVP）的训练框架，兼容u-预测和x₁-预测两种变体，避免显式雅可比计算，大幅降低内存与计算开销。在图像合成、粒子基几何生成及功能生成任务上的实验表明，在固定采样预算下，优化稳定性与样本质量均优于现有方法，相比现有一步生成方法，训练时间与内存消耗减少约50%。
原文摘要 · Abstract (English)
We propose \emph{Euler Mean Flows (EMF)}, a flow-based generative framework for one-step and few-step generation that enforces long-range trajectory consistency with minimal sampling cost. The key idea of EMF is to replace the trajectory consistency constraint, which is difficult to supervise and optimize over long time scales, with a principled linear surrogate that enables direct data supervision for long-horizon flow-map compositions. We derive this approximation from the semigroup formulation of flow-based models and show that, under mild regularity assumptions, it faithfully approximates the original consistency objective while being substantially easier to optimize. This formulation leads to a unified, JVP-free training framework that supports both $u$-prediction and $x_1$-prediction variants, avoiding explicit Jacobian computations and significantly reducing memory and computational overhead. Experiments on image synthesis, particle-based geometry generation, and functional generation demonstrate improved optimization stability and sample quality under fixed sampling budgets, together with approximately $50\%$ reductions in training time and memory consumption compared to existing one-step methods for image generation.
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