Spectral methods for Neural Integral Equations
Abstract
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.
Cite
@article{arxiv.2312.05654,
title = {Spectral methods for Neural Integral Equations},
author = {Emanuele Zappala},
journal= {arXiv preprint arXiv:2312.05654},
year = {2025}
}
Comments
v4: There were various inaccuracies that have been fixed. First, the approach was performed on a proper subspace of X (H\"older space), rather than on X as declared at some point. Second, in the preamble to Theorem 3.5 a hypothesis was not listed: P_N's are assumed to be uniformly bounded on the space R, to show that P_N approximates the identity on R, rather than the opposite at it was written