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：神经积分方程是基于积分方程理论的深度学习模型，该模型包含一个积分算子及其对应的第二类积分方程，并通过优化过程进行学习。这种方法能够利用积分算子在机器学习中的非局部特性，但计算成本较高。本文提出了一种基于谱方法的神经积分方程框架，可在谱域中学习算子，从而降低计算成本并实现高插值精度。我们研究了所提方法的性质，展示了模型逼近能力的理论保障以及数值方法解的收敛性，并通过数值实验验证了模型的实际有效性。