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 can SPM 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仍是一项巨大挑战。本文基于Lawson-Euler格式的弱形式，通过追踪粒子的确定性运动、随机跳跃、重采样和重加权，提出了一种随机粒子方法（SPM）。SPM采用实值加权粒子近似高维解，可自动调整点分布以逼近解的相关特征。为评估非线性项，本文采用基于虚拟均匀网格的分段常数重构方法，充分利用了SPM的内在自适应特性。两者结合使SPM实现了时间上的自适应采样目标。对6维Allen-Cahn方程和7维Hamilton-Jacobi-Bellman方程的数值实验表明，SPM能在保持可接受精度的同时，高效求解高维非线性偏微分方程。