用随机方法求解带漂移和衰减的椭圆方程，精度高且计算复杂度可控。
Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing
- 基于改进的走球算法构造蒙特卡洛估计器，显式利用随机时间信息。
- 误差在精度和维数上均多项式增长，保证了高效性。
- 首次实现深度神经网络对这类方程解的高效逼近，适合高维问题研究者。
本文针对具有常数扩散、漂移和衰减项的线性椭圆型偏微分方程，提供了蒙特卡洛与深度神经网络近似方法。基于Beznea等人提出的改进走球算法（arXiv:2209.01432），我们构建了显式包含随机时间采样的蒙特卡洛估计器，并建立了统一的误差界。在适当假设下，所需样本量随逆精度和问题维度的增长最多为多项式级。此外，我们证明了对随机表示的深度神经网络逼近结果：在边界数据和到边界距离函数可由神经网络良好表示的前提下，所设计的网络参数量同样以逆精度和维数的多项式增长，实现一致逼近。该工作将先前复杂度分析拓展至含漂移与衰减的更广类椭圆方程。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。