In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to H\" older data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, { and that this can be done highly efficiently if we use a modified version} of the walk on spheres algorithm { as an acceleration method. This provides estimates which are efficient with respect to the prescribed approximation error and with polynomial complexity in the dimension and the reciprocal of the error.} {A crucial feature is that} the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders { in a constructive way} ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomialy in the dimension $d$ and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.

翻译：本文研究在$\mathbb{R}^d$的一般有界区域中，具有Hölder数据的泊松方程解的蒙特卡洛与神经网络逼近问题。我们聚焦两个基本目标：首要且最重要的目标是证明，利用球上行走算法的改进版本作为加速方法，可通过蒙特卡洛方法在最大模范数下高效数值逼近泊松方程的解。该方案在预设逼近误差下具有高效的估计性能，其复杂度随维度和误差倒数呈多项式增长。关键特征在于，总体样本数量不依赖于执行逼近的空间点位置。第二个目标是证明，所获得的蒙特卡洛求解器能以构造性方式生成泊松问题的ReLU深度神经网络解，其网络规模关于维度$d$和期望误差的依赖关系至多为多项式级。事实上我们证明，随机深度神经网络能以高概率实现较小的逼近误差，并具有低维数多项式复杂度。