In order to numerically solve high-dimensional nonlinear PDEs and alleviate the curse of dimensionality, a stochastic particle method (SPM) has been proposed to capture the relevant feature of the solution through the adaptive evolution of particles [J. Comput. Phys. 527 (2025) 113818]. In this paper, we introduce an active birth-death dynamics of particles to improve the efficiency of SPM. The resulting method, dubbed SPM-birth-death, sample new particles according to the nonlinear term and execute the annihilation strategy when the number of particles exceeds a given threshold. Preliminary numerical experiments on the Allen-Cahn equation demonstrate that SPM-birth-death can achieve smaller errors at the same computational cost compared with the original SPM.

翻译：为数值求解高维非线性偏微分方程并缓解维度灾难，已有研究提出通过粒子的自适应演化来捕捉解相关特征的随机粒子方法[J. Comput. Phys. 527 (2025) 113818]。本文引入主动的粒子生灭动力学以提升随机粒子方法的效率。所提出的方法（称为SPM-birth-death）依据非线性项生成新粒子，并在粒子数量超过给定阈值时执行湮灭策略。针对Allen-Cahn方程的初步数值实验表明，在相同计算成本下，SPM-birth-death相比原始随机粒子方法可获得更小的误差。