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Open AccessArticle
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations
by
Shanhao Yuan
Shanhao Yuan
Yanqin Liu
Yanqin Liu
Limei Yan
Limei Yan
Runfa Zhang
Runfa Zhang
and
Shunjun Wu
Shunjun Wu
1
School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China
2
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
3
School of Automation and Software Engineering, Shanxi University, Taiyuan 030013, China
4
Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(8), 541; https://doi.org/10.3390/fractalfract9080541 (registering DOI)
Submission received: 16 July 2025
/
Revised: 5 August 2025
/
Accepted: 12 August 2025
/
Published: 16 August 2025
Abstract
This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields.
Share and Cite
MDPI and ACS Style
Yuan, S.; Liu, Y.; Yan, L.; Zhang, R.; Wu, S.
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations. Fractal Fract. 2025, 9, 541.
https://doi.org/10.3390/fractalfract9080541
AMA Style
Yuan S, Liu Y, Yan L, Zhang R, Wu S.
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations. Fractal and Fractional. 2025; 9(8):541.
https://doi.org/10.3390/fractalfract9080541
Chicago/Turabian Style
Yuan, Shanhao, Yanqin Liu, Limei Yan, Runfa Zhang, and Shunjun Wu.
2025. "Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations" Fractal and Fractional 9, no. 8: 541.
https://doi.org/10.3390/fractalfract9080541
APA Style
Yuan, S., Liu, Y., Yan, L., Zhang, R., & Wu, S.
(2025). Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations. Fractal and Fractional, 9(8), 541.
https://doi.org/10.3390/fractalfract9080541
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