In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder 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：本文研究一般有界区域 $\mathbb{R}^d$ 上满足 Hölder 数据的泊松方程解的概率逼近与神经网络逼近。我们致力于实现两个基本目标。首要且最关键的是，我们证明了泊松方程的解可通过蒙特卡洛方法在 sup-范数意义下进行数值逼近，并且若采用修正版的球面行走算法作为加速方法，此过程可实现高效计算。该方法提供的估计在给定逼近误差下具有高效性，且在维度与误差倒数的关系上呈现多项式复杂度。一个重要特征是，所需总样本量不依赖于进行逼近的点位。作为第二个目标，我们证明了所获得的蒙特卡洛求解器能以构造性方式得到泊松问题的 ReLU 深度神经网络解，其网络规模至多随维度 $d$ 与目标误差呈多项式依赖。事实上，我们证明该随机深度神经网络能以高概率实现较小的逼近误差，并在维度上具有低次多项式复杂度。