The problem of approximating the solution operator of a PDE (see as a mapping between infinite dimensional spaces) is of great importance in science and engineering, whenever an application requires multiple solutions of similar problems. In recent years, several techniques based on neural networks (NN) and polynomials have been developed to tackle this issue.

In this talk, I will present some theoretical, quantitative results on the approximation of solution operators of some elliptic PDEs by NN and polynomial surrogates. I will discuss the convergence rates of neural operators and how they depend on the smoothness of the coefficients in the input sets. I will consider PDEs in smooth and non-smooth domains, with bounded coefficients and with coefficients that are distributed log-normally.