  Many applications, such as system identification, classification of time
series, direct and inverse problems in partial differential equations, and
uncertainty quantification lead to the question of approximation of a
non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We
study the problem of determining the degree of approximation of such operators
on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount
of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well
established strategy to approximate $\mathcal{F}(F)$ for some $F\in
K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a
finite number $d$ (repectively $m$) of real numbers. Together with appropriate
reconstruction algorithms (decoders), the problem reduces to the approximation
of $m$ functions on a compact subset of a high dimensional Euclidean space
$\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in
$\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the
complexity of the approximation on $\mathbb{S}^d$ are all large, and it is
necessary to estimate the accuracy keeping track of the inter-dependence of all
the approximations involved. In this paper, we establish constructive methods
to do this efficiently; i.e., with the constants involved in the estimates on
the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study
different smoothness classes for the operators, and also propose a method for
approximation of $\mathcal{F}(F)$ using only information in a small
neighborhood of $F$, resulting in an effective reduction in the number of
parameters involved.
