Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model
This study utilizes novel Kolmogorov-Arnold Networks to solve the fifth-order KdV-Lax problem, employing both periodic and Peakon solutions. Several soliton solutions, including solitary wave, Peakon forms, are presented using the KANs technique for the Lax problem of fifth order. The novelty of thi...
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| Language: | English |
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Elsevier
2025-09-01
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| Series: | Case Studies in Thermal Engineering |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2214157X2500797X |
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| author | Muhammad Wajahat Anjum Sana Ullah Saqib Yin-Tzer Shih Israr Ul Hassan Adnan Ines Hilali Jaghdam Mouloud Aoudia Lioua Kolsi |
| author_facet | Muhammad Wajahat Anjum Sana Ullah Saqib Yin-Tzer Shih Israr Ul Hassan Adnan Ines Hilali Jaghdam Mouloud Aoudia Lioua Kolsi |
| author_sort | Muhammad Wajahat Anjum |
| collection | DOAJ |
| description | This study utilizes novel Kolmogorov-Arnold Networks to solve the fifth-order KdV-Lax problem, employing both periodic and Peakon solutions. Several soliton solutions, including solitary wave, Peakon forms, are presented using the KANs technique for the Lax problem of fifth order. The novelty of this investigation lies in its combination of a practical numerical approach for KANs with a powerful analytical strategy (the exp-function method) to verify reliability and authenticity, utilizing illustrations and tables to identify various types of soliton solutions. Although KANs provide an approximation for solving the KdV-Lax equation, their ability to tackle complex problems makes them a viable choice when an analytical solution is not possible. The use of KANs over the exponential function method's approximate solution is exceptional; it offers a thorough analysis of the solution's uniqueness and convergence using loss plots, error histograms, mean square logarithmic error, mean poisson deviation, and regression plots, among others. The numerical results of thorough simulations, with minimal error (MLSE≤10−6), regression metric value (D2≈1), and histogram with the majority of instances (≥85%) nearly close to zero, unquestionably supported or confirmed the accuracy of KANs against the analytical (exp-function) method. Furthermore, the robustness of the KANs model is demonstrated by consistently low absolute error metrics. |
| format | Article |
| id | doaj-art-9b1e0bc9bd074d36a255a2fe7e5bcd4e |
| institution | Kabale University |
| issn | 2214-157X |
| language | English |
| publishDate | 2025-09-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Case Studies in Thermal Engineering |
| spelling | doaj-art-9b1e0bc9bd074d36a255a2fe7e5bcd4e2025-08-20T03:26:51ZengElsevierCase Studies in Thermal Engineering2214-157X2025-09-017310653710.1016/j.csite.2025.106537Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax modelMuhammad Wajahat Anjum0Sana Ullah Saqib1Yin-Tzer Shih2Israr Ul Hassan3 Adnan4Ines Hilali Jaghdam5Mouloud Aoudia6Lioua Kolsi7Department of Mathematics, COMSATS University Islamabad, Attock Campus, PakistanDepartment of Applied Mathematics, National Chung Hsing University, Taichung, TaiwanDepartment of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan; Corresponding author.Department of Electrical Engineering, COMSATS University Islamabad, Attock Campus, PakistanDepartment of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif, AJ&K, 12080, Pakistan; Corresponding author.Department of Computer Science and Information Technology, Applied College, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Industrial Engineering, College of Engineering, Northern Border University, P.O. Box 1321, Arar, 91431, Saudi ArabiaDepartment of Mechanical Engineering, College of Engineering, University of Ha'il, Ha'il City, 81451, Saudi ArabiaThis study utilizes novel Kolmogorov-Arnold Networks to solve the fifth-order KdV-Lax problem, employing both periodic and Peakon solutions. Several soliton solutions, including solitary wave, Peakon forms, are presented using the KANs technique for the Lax problem of fifth order. The novelty of this investigation lies in its combination of a practical numerical approach for KANs with a powerful analytical strategy (the exp-function method) to verify reliability and authenticity, utilizing illustrations and tables to identify various types of soliton solutions. Although KANs provide an approximation for solving the KdV-Lax equation, their ability to tackle complex problems makes them a viable choice when an analytical solution is not possible. The use of KANs over the exponential function method's approximate solution is exceptional; it offers a thorough analysis of the solution's uniqueness and convergence using loss plots, error histograms, mean square logarithmic error, mean poisson deviation, and regression plots, among others. The numerical results of thorough simulations, with minimal error (MLSE≤10−6), regression metric value (D2≈1), and histogram with the majority of instances (≥85%) nearly close to zero, unquestionably supported or confirmed the accuracy of KANs against the analytical (exp-function) method. Furthermore, the robustness of the KANs model is demonstrated by consistently low absolute error metrics.http://www.sciencedirect.com/science/article/pii/S2214157X2500797XKolmogorov-Arnold networksIntelligent computingLax equationPeriodic and Peakon solutions |
| spellingShingle | Muhammad Wajahat Anjum Sana Ullah Saqib Yin-Tzer Shih Israr Ul Hassan Adnan Ines Hilali Jaghdam Mouloud Aoudia Lioua Kolsi Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model Case Studies in Thermal Engineering Kolmogorov-Arnold networks Intelligent computing Lax equation Periodic and Peakon solutions |
| title | Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model |
| title_full | Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model |
| title_fullStr | Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model |
| title_full_unstemmed | Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model |
| title_short | Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model |
| title_sort | application of kolmogorov arnold network kan for solitary peakon investigation of lax model |
| topic | Kolmogorov-Arnold networks Intelligent computing Lax equation Periodic and Peakon solutions |
| url | http://www.sciencedirect.com/science/article/pii/S2214157X2500797X |
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