  Operator learning has emerged as a new paradigm for the data-driven
approximation of nonlinear operators. Despite its empirical success, the
theoretical underpinnings governing the conditions for efficient operator
learning remain incomplete. The present work develops theory to study the data
complexity of operator learning, complementing existing research on the
parametric complexity. We investigate the fundamental question: How many
input/output samples are needed in operator learning to achieve a desired
accuracy $\epsilon$? This question is addressed from the point of view of
$n$-widths, and this work makes two key contributions. The first contribution
is to derive lower bounds on $n$-widths for general classes of Lipschitz and
Fr\'echet differentiable operators. These bounds rigorously demonstrate a
``curse of data-complexity'', revealing that learning on such general classes
requires a sample size exponential in the inverse of the desired accuracy
$\epsilon$. The second contribution of this work is to show that ``parametric
efficiency'' implies ``data efficiency''; using the Fourier neural operator
(FNO) as a case study, we show rigorously that on a narrower class of
operators, efficiently approximated by FNO in terms of the number of tunable
parameters, efficient operator learning is attainable in data complexity as
well. Specifically, we show that if only an algebraically increasing number of
tunable parameters is needed to reach a desired approximation accuracy, then an
algebraically bounded number of data samples is also sufficient to achieve the
same accuracy.
