The Random Batch Method (RBM) is an effective technique to reduce the computational complexity when solving certain stochastic differential problems (SDEs) involving interacting particles. It can transform the computational complexity from O(N^2) to O(N), where N represents the number of particles. However, the traditional RBM can only be effectively applied to interacting particle systems with relatively smooth kernel functions to achieve satisfactory results. To address the issue of non-convergence of the RBM in particle interaction systems with significant singularities, we propose some enhanced methods to make the modified algorithm more applicable. The idea for improvement primarily revolves around a momentum-like correction, and we refer to the enhanced algorithm as the Random Batch Method with Momentum Correction ( RBM-M). We provide a theoretical proof to control the error of the algorithm, which ensures that under ideal conditions it has a smaller error than the original algorithm. Finally, numerical experiments have demonstrated the effectiveness of the RBM-M algorithm.

翻译：随机批处理方法（RBM）是求解涉及相互作用粒子的某些随机微分问题（SDEs）时，降低计算复杂度的一种有效技术。它可以将计算复杂度从 O(N^2) 降低至 O(N)，其中 N 表示粒子数量。然而，传统的 RBM 只能有效地应用于具有相对平滑核函数的相互作用粒子系统，以获得令人满意的结果。为了解决 RBM 在具有显著奇异性的粒子相互作用系统中的非收敛性问题，我们提出了一些增强方法，以使改进后的算法更具适用性。改进的思路主要围绕一种类动量的修正展开，我们将增强后的算法称为带动量修正的随机批处理方法（RBM-M）。我们提供了控制算法误差的理论证明，该证明确保了在理想条件下，其误差小于原始算法。最后，数值实验验证了 RBM-M 算法的有效性。