Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

翻译：高维非线性偏微分方程的数值求解因维数灾难问题而始终面临巨大挑战。本文从Lawson-Euler格式的弱形式出发，通过追踪粒子的确定性运动、随机跳跃、重采样与重加权过程，提出了一种随机粒子方法。该方法采用实值加权粒子来逼近高维解，其粒子点分布可自动调整以捕捉解的相关特征。通过引入虚拟均匀网格进行分段常数重构来评估非线性项，从而充分利用了SPM固有的自适应特性。二者结合使得SPM能够实现时间维度上的自适应采样。针对6维Allen-Cahn方程与7维Hamilton-Jacobi-Bellman方程的数值实验表明，SPM在保持可接受精度的同时，能够高效求解高维非线性偏微分方程，展现出其应用潜力。