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.

翻译：受通信或计算约束限制，大规模群体通常仅能通过有限次控制更新进行引导，而现有的大多数学习方法并未考虑这一事实，转而建模瞬时速度场。因此，决策的自然对象应是有限窗口控制量而非无穷小量。为弥补这一空白，我们引入近期机器学习框架平均流（MeanFlow）并将其推广至具有一般线性动力系统的场景，从而构建一种直接在控制空间运作的新型采样数据学习框架，可应用于群体引导任务。具体而言，我们学习参数化各区间最小能量控制输入的有限时域系数，并推导出将该系数与局部桥诱导监督信号相关联的微分恒等式。该恒等式导出一个简单的停止梯度回归目标，使得区间系数场可通过桥样本高效学习。所学习策略通过采样数据更新实施，确保最终控制器严格遵循预设的线性时不变动力学与驱动通道特性。该方法在保持底层控制系统有限窗口驱动结构一致性的前提下，实现了大规模群体的少步引导。