  Neural integral equations are deep learning models based on the theory of
integral equations, where the model consists of an integral operator and the
corresponding equation (of the second kind) which is learned through an
optimization procedure. This approach allows to leverage the nonlocal
properties of integral operators in machine learning, but it is computationally
expensive. In this article, we introduce a framework for neural integral
equations based on spectral methods that allows us to learn an operator in the
spectral domain, resulting in a cheaper computational cost, as well as in high
interpolation accuracy. We study the properties of our methods and show various
theoretical guarantees regarding the approximation capabilities of the model,
and convergence to solutions of the numerical methods. We provide numerical
experiments to demonstrate the practical effectiveness of the resulting model.
