Is Flow Matching Just Trajectory Replay for Sequential Data?
This work clarifies the theoretical underpinnings of flow matching for sequential data, offering interpretable insights and improved sampling methods for time-series generation.
The paper investigates whether flow matching for time-series generation learns general dynamics or merely replays trajectories, showing that the optimal velocity field is a similarity-weighted mixture of past transitions. The derived closed-form sampler achieves strong probabilistic forecasts on nonlinear dynamical system benchmarks without training.
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.