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.
This repository provides a minimal implementation of FreeFM, a flow matching inspired training-free model for probabilistic forecasting.
First, install packages from requirements.txt. Then we can run the code with simple examples below.
python main.py --system dho --mode topk --device cuda:0 --integrator euler --ode_steps 100 --topk 64
python main.py --system lorenz63 --mode topk --device cuda:0 --integrator euler --ode_steps 100 --topk 256
If you find our work useful for your research, please consider citing our paper:
@article{lim2026flow,
title={Is Flow Matching Just Trajectory Replay for Sequential Data?},
author={Lim, Soon Hoe and Lin, Shizheng and Mahoney, Michael W and Erichson, N Benjamin},
journal={arXiv preprint arXiv:2602.08318},
year={2026}
}