Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.

翻译：关于平均场相互作用粒子系统的随机批方法（RBM）能够将与粒子相互作用相关的二次计算成本降低至近线性成本。这类算法的核心在于在每个时间步对粒子系综进行随机划分，形成较小的批。每个批内粒子的相互作用随后演化至下一个时间步。该方法在将计算成本降低一个数量级的同时，也因随机划分而增加了波动幅度。本研究针对福克-普朗克型非局部偏微分方程中的RBM，提出一种基于控制变量策略的方差缩减技术。其核心思想是构建一个替代模型，该模型能以线性成本对全体粒子进行计算，同时保持与原粒子动力学的高度相关性。意见传播和群体运动等集体行为模型中的实例证明了该方法具有巨大潜力。