FlowCTS: On-policy Continuous Trajectory Supervision of Flow Models

While on-policy distillation (OPD) effectively addresses sparse rewards and exposure bias in large language model post-training, its extension to flow models remains underexplored. To this end, we propose Flow Continuous Trajectory Supervision (FlowCTS), which matches subsequent student and reference trajectories initialized from the same student-visited state. Using the integral relation between trajectories and velocity fields, we derive a temporally weighted velocity-matching upper bound and discretize it into practical objectives parameterized by the number of supervision steps. Under a multi-reference setup, FlowCTS-OPD outperforms vanilla KL-based OPD with faster convergence. Compared with the SD3.5-M base model, it improves GenEval from $0.63$ to $0.93$, OCR from $0.59$ to $0.92$, and PickScore from $21.72$ to $23.06$, matching task experts and surpassing the GenEval expert under multi-step supervision, while outperforming mixed-reward RL across all metrics. Further analysis reveals a clear temporal supervision mismatch in vanilla KL-based OPD arising from its auxiliary SDE transition kernels. Beyond on-policy setting, FlowCTS also consistently outperforms vanilla SFT especially on OCR, while increasing supervision steps exhibit a trade-off between richer trajectory information and greater optimization difficulty.