Is Flow Matching Just Trajectory Replay for Sequential Data?

TL;DR

This paper investigates whether flow matching learns transferable dynamical structures or merely replays trajectories, providing a theoretical characterization and a practical sampler called FreeFM for time series generation.

Contribution

It derives the velocity field targeted by flow matching, revealing dataset dependence and proposing a nonparametric interpretation and approximation schemes.

Findings

The implied sampler is an ODE with dataset-dependent dynamics.

The optimal velocity field is a similarity-weighted mixture of observed velocities.

FreeFM provides strong probabilistic forecasts without training.

Abstract

Flow matching (FM) is increasingly used in scientific domains for time series generation and forecasting, where data often arise from underlying dynamical systems. However, it is not well-understood whether it learns transferable 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 observed transitions, making the dataset dependence explicit and interpretable. This characterization positions neural FM models as parametric surrogates of an ideal nonparametric solution and suggests practical approximation schemes for robust ODE-based generation. As a byproduct of our analysis, the resulting closed-form sampler, FreeFM, provides strong probabilistic forecasts on nonlinear dynamical system benchmarks directly from historical transitions, without training.