MeanFlow Meets Control: Scaling Sampled-Data Control for Swarms

Steering large-scale swarms in only a few control updates is challenging because real systems operate in sampled-data form: control inputs are updated intermittently and applied over finite intervals. In this regime, the natural object is not an instantaneous velocity field, but a finite-window control quantity that captures the system response over each sampling interval. Inspired by MeanFlow, we introduce a control-space learning framework for swarm steering under linear time-invariant dynamics. The learned object is the coefficient that parameterizes the finite-horizon minimum-energy control over each interval. We show that this coefficient admits both an integral representation and a local differential identity along bridge trajectories, which leads to a simple stop-gradient training objective. At implementation time, the learned coefficient is used directly in sampled-data updates, so the prescribed dynamics and actuation map are respected by construction. The resulting framework provides a scalable approach to few-step swarm steering that is consistent with the sampled-data structure of real control systems.