🤖 AI Summary
This work proposes an efficient method for solving linear elliptic partial differential equations with constant diffusion, drift, and killing terms by integrating an enhanced Walk-on-Spheres (WoS) Monte Carlo algorithm with deep neural networks. The approach constructs unbiased estimators through explicit stochastic time sampling and employs a tailored neural network architecture to approximate both the stochastic representation of the solution and the boundary data. It represents the first integration of stochastic representations for elliptic PDEs with drift and killing terms into a deep learning framework. The study establishes uniform error bounds for the Monte Carlo estimator and proves that the solution approximation achieves polynomial complexity in both accuracy and dimensionality, thereby significantly extending the theoretical foundations and practical applicability of numerical methods for high-dimensional PDEs.
📝 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.