The random batch method (RBM) proposed in [Jin et al., J. Comput. Phys., 400(2020), 108877] for large interacting particle systems is an efficient with linear complexity in particle numbers and highly scalable algorithm for $N$-particle interacting systems and their mean-field limits when $N$ is large. We consider in this work the quantitative error estimate of RBM toward its mean-field limit, the Fokker-Planck equation. Under mild assumptions, we obtain a uniform-in-time $O(\tau^2 + 1/N)$ bound on the scaled relative entropy between the joint law of the random batch particles and the tensorized law at the mean-field limit, where $\tau$ is the time step size and $N$ is the number of particles. Therefore, we improve the existing rate in discretization step size from $O(\sqrt{\tau})$ to $O(\tau)$ in terms of the Wasserstein distance.

翻译：随机批处理方法（RBM）由Jin等人提出[J. Comput. Phys., 400(2020), 108877]，适用于大规模相互作用粒子系统。该方法在处理$N$粒子相互作用系统及其在大$N$条件下的平均场极限时，具有线性复杂度且高度可扩展。本文研究了RBM在其平均场极限（即Fokker-Planck方程）下的定量误差估计。在温和假设下，我们得到了关于随机批处理粒子联合分布与平均场极限处张量化分布之间的尺度化相对熵的一致的$O(\tau^2 + 1/N)$时间界，其中$\tau$为时间步长，$N$为粒子数量。由此，在Wasserstein距离度量下，我们将现有离散步长误差率从$O(\sqrt{\tau})$改进至$O(\tau)$。