Iulian Cîmpean
University of Bucharest & Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

We present probabilistic and neural network approximations for solutions to Poisson equation subject to Holder continuous Dirichlet boundary conditions in general bounded domains in $\mathbb{R}^d$. Our main results are two-folded: On the one hand we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, without the curse of high dimensions and efficiently with respect to the prescribed approximation error. On the other hand, we show that the obtained Monte Carlo solver renders a random ReLU deep neural network (DNN) that provides with high probability a small approximation error and low polynomial complexity in the dimension. This is joint work with L. Beznea, O. Lupaşcu, I. Popescu, and A. D. Zărnescu.