# Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations

arXiv: https://arxiv.org/abs/2401.07003 · HTML: https://mlsanity.com/paper/2401.07003/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 13 Jan 2024.

## Builds on (most cited first, 9 of 9)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Convergence of Deep ReLU Networks](https://mlsanity.com/paper/2107.12530/) (2021)
- [Convergence of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2109.13542/) (2021)
- [Multi-Grade Deep Learning](https://mlsanity.com/paper/2302.00150/) (2023)
- [Sparse Deep Neural Network for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2207.13266/) (2022)
- [Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation](https://mlsanity.com/paper/2309.07401/) (2023)

## Built on by (most cited first, 5 of 5)

- [Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learning](https://mlsanity.com/paper/2410.16105/) (2024)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)
- [Computational Advantages of Multi-Grade Deep Learning: Convergence Analysis and Performance Insights](https://mlsanity.com/paper/2507.20351/) (2025)
- [Multigrade Neural Network Approximation](https://mlsanity.com/paper/2601.16884/) (2026)
- [Parametric Numerical Integration with (Differential) Machine Learning](https://mlsanity.com/paper/2512.11530/) (2025)

## Abstract (arXiv)

We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favour low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multi-grade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers.