Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization
In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the applic...
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| Format: | Article |
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MDPI AG
2025-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/4/225 |
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| author | Tauqeer Ahmad Muhammad Sulaiman David Bassir Fahad Sameer Alshammari Ghaylen Laouini |
| author_facet | Tauqeer Ahmad Muhammad Sulaiman David Bassir Fahad Sameer Alshammari Ghaylen Laouini |
| author_sort | Tauqeer Ahmad |
| collection | DOAJ |
| description | In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering. |
| format | Article |
| id | doaj-art-4f25b60b7f704bfe9884bc24b5616b22 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-4f25b60b7f704bfe9884bc24b5616b222025-08-20T02:28:14ZengMDPI AGFractal and Fractional2504-31102025-04-019422510.3390/fractalfract9040225Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search OptimizationTauqeer Ahmad0Muhammad Sulaiman1David Bassir2Fahad Sameer Alshammari3Ghaylen Laouini4Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, PakistanDepartment of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, PakistanSmart Structural Health Monitoring and Control Lab (SSHMC Lab), CNAM-Dongguan University of Technology, D1, Daxue Rd., Songshan Lake, Dongguan 523000, ChinaDepartment of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaCollege of Engineering and Technology, American University of the Middle East, Egaila 54200, KuwaitIn this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering.https://www.mdpi.com/2504-3110/9/4/225fractional partial differential equationsphysics-informed neural networksMonte Carlo methodscuckoo search algorithmnumerical analysiscomputational efficiency |
| spellingShingle | Tauqeer Ahmad Muhammad Sulaiman David Bassir Fahad Sameer Alshammari Ghaylen Laouini Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization Fractal and Fractional fractional partial differential equations physics-informed neural networks Monte Carlo methods cuckoo search algorithm numerical analysis computational efficiency |
| title | Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization |
| title_full | Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization |
| title_fullStr | Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization |
| title_full_unstemmed | Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization |
| title_short | Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization |
| title_sort | enhanced numerical solutions for fractional pdes using monte carlo pinns coupled with cuckoo search optimization |
| topic | fractional partial differential equations physics-informed neural networks Monte Carlo methods cuckoo search algorithm numerical analysis computational efficiency |
| url | https://www.mdpi.com/2504-3110/9/4/225 |
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