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

From Monte Carlo to neural networks approximations of boundary value problems