Steering large-scale swarms with only limited control updates is often needed due to communication or computational constraints, yet most learning-based approaches do not account for this and instead model instantaneous velocity fields. As a result, the natural object for decision making is a finite-window control quantity rather than an infinitesimal one. To address this gap, we consider the recent machine learning framework MeanFlow and generalize it to the setting with general linear dynamic systems. This results in a new sampled-data learning framework that operates directly in control space and that can be applied for swarm steering. To this end, we learn the finite-horizon coefficient that parameterizes the minimum-energy control applied over each interval, and derive a differential identity that connects this quantity to a local bridge-induced supervision signal. This identity leads to a simple stop-gradient regression objective, allowing the interval coefficient field to be learned efficiently from bridge samples. The learned policy is deployed through sampled-data updates, guaranteeing that the resulting controller exactly respects the prescribed linear time-invariant dynamics and actuation channel. The resulting method enables few-step swarm steering at scale, while remaining consistent with the finite-window actuation structure of the underlying control system.

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