  Fourier neural operators (FNOs) have recently been proposed as an effective
framework for learning operators that map between infinite-dimensional spaces.
We prove that FNOs are universal, in the sense that they can approximate any
continuous operator to desired accuracy. Moreover, we suggest a mechanism by
which FNOs can approximate operators associated with PDEs efficiently. Explicit
error bounds are derived to show that the size of the FNO, approximating
operators associated with a Darcy type elliptic PDE and with the incompressible
Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in
terms of the reciprocal of the error. Thus, FNOs are shown to efficiently
approximate operators arising in a large class of PDEs.
