  In this paper we investigate the use of Fourier Neural Operators (FNOs) for
image classification in comparison to standard Convolutional Neural Networks
(CNNs). Neural operators are a discretization-invariant generalization of
neural networks to approximate operators between infinite dimensional function
spaces. FNOs - which are neural operators with a specific parametrization -
have been applied successfully in the context of parametric PDEs. We derive the
FNO architecture as an example for continuous and Fr\'echet-differentiable
neural operators on Lebesgue spaces. We further show how CNNs can be converted
into FNOs and vice versa and propose an interpolation-equivariant adaptation of
the architecture.
