Path Guiding for Monte Carlo PDE Solvers

Tianyu Huang, Jingwang Ling, Shuang Zhao, Feng Xu

Abstract: Walk on stars (WoSt) has shown its power in being applied to Monte Carlo methods for solving partial differential equations, but the sampling techniques in WoSt are not satisfactory, leading to high variance. We propose a guiding-based importance sampling method to reduce the variance of WoSt. Drawing inspiration from path guiding in rendering, we approximate the directional distribution of the recursive term of WoSt using online-learned parametric mixture distributions, decoded by a lightweight neural field. This adaptive approach enables importance sampling the recursive term, which lacks shape information before computation. We introduce a reflection technique to represent guiding distributions at Neumann boundaries and incorporate multiple importance sampling with learnable selection probabilities to further reduce variance. We also present a practical GPU implementation of our method. Experiments show that our method effectively reduces variance compared to the original WoSt, given the same time or the same sample budget. Code and data will be released.