Document Type

Article

Publication Date

2026

DOI

10.1007/s10915-026-03189-9

Publication Title

Journal of Scientific Computing

Volume

106

Issue

3

Pages

64 (1-33)

Abstract

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

Rights

© The Authors 2026

This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Data Availability

Article states: "The data will be made available on reasonable request."

Original Publication Citation

Jiang, J., & Xu, Y. (2026). Adaptive multi-grade deep learning for highly oscillatory Fredholm integral equations of the second kind. Journal of Scientific Computing, 106(3), 1-33, Article 64. https://doi.org/10.1007/s10915-026-03189-9

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