Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing
2026-08-10 • Machine Learning
Machine Learning
AI summaryⓘ
The authors develop new ways to estimate solutions to certain math problems called linear elliptic partial differential equations, which can model things like heat or diffusion. They improve a method called the Walk-on-Spheres algorithm by using random sampling techniques and deep neural networks. Their approach ensures good accuracy with a manageable amount of computation, even as the problem gets more complex. They also prove that neural networks can approximate these solutions efficiently under certain conditions.
Monte Carlo methodsDeep neural networksElliptic partial differential equationsWalk-on-Spheres algorithmStochastic representationsDriftKilling termError boundsSample complexityBoundary conditions
Authors
Konrad Kleinberg, Thomas Kruse
Abstract
In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.