  Neural operator architectures approximate operators between
infinite-dimensional Banach spaces of functions. They are gaining increased
attention in computational science and engineering, due to their potential both
to accelerate traditional numerical methods and to enable data-driven
discovery. As the field is in its infancy basic questions about minimal
requirements for universal approximation remain open. It is clear that any
general approximation of operators between spaces of functions must be both
nonlocal and nonlinear. In this paper we describe how these two attributes may
be combined in a simple way to deduce universal approximation. In so doing we
unify the analysis of a wide range of neural operator architectures and open up
consideration of new ones.
  A popular variant of neural operators is the Fourier neural operator (FNO).
Previous analysis proving universal operator approximation theorems for FNOs
resorts to use of an unbounded number of Fourier modes, relying on intuition
from traditional analysis of spectral methods. The present work challenges this
point of view: (i) the work reduces FNO to its core essence, resulting in a
minimal architecture termed the ``averaging neural operator'' (ANO); and (ii)
analysis of the ANO shows that even this minimal ANO architecture benefits from
universal approximation. This result is obtained based on only a spatial
average as its only nonlocal ingredient (corresponding to retaining only a
\emph{single} Fourier mode in the special case of the FNO). The analysis paves
the way for a more systematic exploration of nonlocality, both through the
development of new operator learning architectures and the analysis of existing
and new architectures. Numerical results are presented which give insight into
complexity issues related to the roles of channel width (embedding dimension)
and number of Fourier modes.
