Guiding-Based Importance Sampling for Walk on Stars

2024-10-24

AI Summary

WoSt methods for solving partial differential equations suffer from high estimation variance due to poor sampling quality—particularly in complex geometries and near Neumann boundaries. To address this, we propose a guided adaptive importance sampling framework: (i) we introduce path-guiding principles from rendering into WoSt for the first time; (ii) we design a lightweight online-learned neural field that parameterizes a recursive-term mixture distribution without requiring prior geometric knowledge; and (iii) we formulate a reflection-aware Neumann boundary representation, coupled with a learnable multi-importance sampling (MIS) mechanism. Our approach significantly reduces variance under identical time or sample budgets, while enabling efficient GPU parallelization. The method demonstrates robust performance across diverse PDEs and domain geometries. Code and datasets will be publicly released.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Walk on stars (WoSt) method
• sampling accuracy
• complex domain and boundary problems

Innovation

• Guided Importance Sampling
• Neural Network Optimization
• Reflective Multi-Sampling Strategy