# Spectral methods for Neural Integral Equations

arXiv: https://arxiv.org/abs/2312.05654 · HTML: https://mlsanity.com/paper/2312.05654/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 39 references that are in the map.
- First posted on arXiv 9 Dec 2023.

## Builds on (most cited first, 6 of 6)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Neural Integro-Differential Equations](https://mlsanity.com/paper/2206.14282/) (2022)

## Built on by (most cited first, 2 of 2)

- [Projection Methods for Operator Learning and Universal Approximation](https://mlsanity.com/paper/2406.12264/) (2024)
- [Leray-Schauder Mappings for Operator Learning](https://mlsanity.com/paper/2410.01746/) (2024)

## Abstract (arXiv)

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.