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.

翻译：神经积分方程是一类基于积分方程理论的深度学习模型，该模型由积分算子及相应的第二类积分方程构成，并通过优化过程进行学习。该方法能够利用积分算子在机器学习中的非局部特性，但计算成本高昂。本文提出一种基于谱方法的神经积分方程框架，允许我们在谱域中学习算子，从而降低计算成本并提高插值精度。我们研究了该方法的相关性质，并证明了模型在逼近能力方面的理论保证以及数值方法的解收敛性。通过数值实验验证了该模型的实际有效性。