发表机构
University of Wuppertal（伍珀塔尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于改进球面行走算法，提出带漂移和杀伤项的椭圆型PDE解的蒙特卡罗估计量与深度神经网络近似方法，证明其样本复杂度和参数规模均为多项式级，扩展了相关复杂度分析的适用范围。
AI 中文摘要
本文针对带有常数扩散项、漂移项和杀伤项的线性椭圆型偏微分方程解的随机表示，提供了蒙特卡罗方法和深度神经网络近似方案。基于Beznea等人（arXiv:2209.01432）提出的改进球面行走算法，我们引入了蒙特卡罗估计量，该估计量明确整合了所分析随机表示中产生的采样随机时间。我们为这些估计量建立了一致误差界，并证明在适当假设下，若要达到规定的近似精度，样本复杂度的增长速度最多为精度倒数与问题维度的多项式级。此外，我们还针对该随机表示证明了深度神经网络近似结果：假设边界数据和边界距离函数有合适的神经网络表示，我们利用所构建的蒙特卡罗方法设计深度神经网络，使其以参数数量最多为精度倒数与问题维度的多项式级的规模，一致近似该随机表示。这些结果将此前的复杂度分析扩展至包含漂移项和杀伤项的更广泛椭圆型方程类别。
英文摘要
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.
