Quantitative approximation theorem of neural operators for the Dirichlet-to-Neumann map

In this paper, we study neural operators for the Dirichet-to-Neumann map (DN map), which is encountered in the famous Calderón's problem. In many numerical methods used for reconstructing the unknown conductivity from the knowledge of the DN map need to evaluate the corresponding DN map at a given conductivity function, which involves solving the boundary value problem for the conductivity equation and taking the Neumann data. Such procedure is time consuming and not effective. Hence, it is a favorable strategy to construct a surrogate to replace the forward map mapping conductivity functions to corresponding DN maps. The main theme of this paper is to derive quantitative bounds on the parametric complexity of these neural operators. In particular, under suitable assumptions, their depth grows at most logarithmically with $\varepsilon^{-1}$, while the relevant spectral and projection ranks grow algebraically. It should be noted that general neural operators suffer from "the curse of parametric dimensionality". This work demonstrates the construction of neural operators for the DN map that does not suffer from such curse.