Flow matching (FM) is increasingly used for time-series generation, but it is not well understood whether it learns a general dynamical structure or simply performs an effective "trajectory replay". We study this question by deriving the velocity field targeted by the empirical FM objective on sequential data, in the limit of perfect function approximation. For the Gaussian conditional paths commonly used in practice, we show that the implied sampler is an ODE whose dynamics constitutes a nonparametric, memory-augmented continuous-time dynamical system. The optimal field admits a closed-form expression as a similarity-weighted mixture of instantaneous velocities induced by past transitions, making the dataset dependence explicit and interpretable. This perspective positions neural FM models trained by stochastic optimization as parametric surrogates of an ideal nonparametric solution. Using the structure of the optimal field, we study sampling and approximation schemes that improve the efficiency and numerical robustness of ODE-based generation. On nonlinear dynamical system benchmarks, the resulting closed-form sampler yields strong probabilistic forecasts directly from historical transitions, without training.

翻译：流匹配（FM）在时间序列生成中的应用日益广泛，但其是否真正学习到了通用的动力学结构，还是仅仅执行了一种有效的“轨迹回放”，目前尚未得到充分理解。我们通过推导在完美函数逼近极限下，经验FM目标在序列数据上所针对的速度场来研究这一问题。针对实践中常用的高斯条件路径，我们证明了隐含的采样器是一个常微分方程（ODE），其动力学构成了一个非参数、记忆增强的连续时间动力系统。该最优速度场具有闭式解，可表示为过去转移所诱导的瞬时速度的相似性加权混合，从而使得数据集的依赖性变得显式且可解释。这一视角将基于随机优化训练的神经FM模型定位为理想非参数解的参数化替代。利用最优场的结构，我们研究了能够提高基于ODE的生成方法的效率与数值鲁棒性的采样与逼近方案。在非线性动力系统基准测试中，由此得到的闭式采样器能够直接从历史转移中产生强大的概率预测，而无需进行训练。