A parallel variable-order iteration algorithm for solving continuous integral equations using deep learning

Zhiyuan Ren; Ruilong Yu; Yi Zeng; Shijie Zhou; Qihe Liu; Zhiyuan Ren; Ruilong Yu; Yi Zeng; Shijie Zhou; Qihe Liu

doi:10.3934/math.20261237

AIMS Mathematics

2026, Volume 11, Issue 9: 31329-31372. doi: 10.3934/math.20261237

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Research article

A parallel variable-order iteration algorithm for solving continuous integral equations using deep learning

1.
School of Computer and Network Security, Chengdu University of Technology, No. 1, Erxianqiao East Third Road, Chenghua District, Chengdu 610059, China
2.
School of Information and Software Engineering, University of Electronic Science and Technology of China, No. 5, Section 2, Jianshe North Road, Chenghua District, Chengdu 610054, China

Received: 14 July 2026 Revised: 15 September 2026 Accepted: 18 September 2026 Published: 23 September 2026
MSC : 65R20, 45B05, 45D05, 68T07, 68Q12

Deep learning has become a key approach for solving integral equations, a central theme in numerical computation. However, existing methods often struggle with precision and efficiency. This study presents the parallel variable-order iteration algorithm (P-VOI), inspired by the variable-order method in the R-AIM algorithm, specifically designed to resolve continuous integral equations using deep learning. The P-VOI algorithm transforms these equations into differential forms by applying the Newton-Leibniz formula. It employs parallel iterative processes, allowing neural networks to approximate the unknown and primitive functions simultaneously. This integration of parallel iteration significantly enhances computational efficiency compared to R-AIM. Additionally, the P-VOI algorithm uses neural networks as trial functions during training, addressing challenges that arise when integral kernels have zeros. This method reduces systematic errors from finite-sum representations, improving solution accuracy. Empirical evaluations demonstrate that the P-VOI algorithm achieves superior convergence accuracy (mean squared error in the range of $ 10^{-7} $ to $ 10^{-9} $) and efficiency, with speedups of 2.9× to 5.2× over R-AIM. Compared with non-variable-order methods, this corresponds to an improvement of approximately three orders of magnitude in accuracy based on MSE; relative to the direct variable-order baseline R-AIM, the accuracy remains of the same order of magnitude with slightly higher precision.
- deep learning,
- continuous integral equations,
- parallel iteration,
- variable-order transformation,
- speedup ratio
Citation: Zhiyuan Ren, Ruilong Yu, Yi Zeng, Shijie Zhou, Qihe Liu. A parallel variable-order iteration algorithm for solving continuous integral equations using deep learning[J]. AIMS Mathematics, 2026, 11(9): 31329-31372. doi: 10.3934/math.20261237

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Abstract

Deep learning has become a key approach for solving integral equations, a central theme in numerical computation. However, existing methods often struggle with precision and efficiency. This study presents the parallel variable-order iteration algorithm (P-VOI), inspired by the variable-order method in the R-AIM algorithm, specifically designed to resolve continuous integral equations using deep learning. The P-VOI algorithm transforms these equations into differential forms by applying the Newton-Leibniz formula. It employs parallel iterative processes, allowing neural networks to approximate the unknown and primitive functions simultaneously. This integration of parallel iteration significantly enhances computational efficiency compared to R-AIM. Additionally, the P-VOI algorithm uses neural networks as trial functions during training, addressing challenges that arise when integral kernels have zeros. This method reduces systematic errors from finite-sum representations, improving solution accuracy. Empirical evaluations demonstrate that the P-VOI algorithm achieves superior convergence accuracy (mean squared error in the range of $ 10^{-7} $ to $ 10^{-9} $) and efficiency, with speedups of 2.9× to 5.2× over R-AIM. Compared with non-variable-order methods, this corresponds to an improvement of approximately three orders of magnitude in accuracy based on MSE; relative to the direct variable-order baseline R-AIM, the accuracy remains of the same order of magnitude with slightly higher precision.

References

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