Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods
This study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-12-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/12/872 |
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| author | Svetislav Savović Miloš Ivanović Branko Drljača Ana Simović |
| author_facet | Svetislav Savović Miloš Ivanović Branko Drljača Ana Simović |
| author_sort | Svetislav Savović |
| collection | DOAJ |
| description | This study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated to demonstrate the effectiveness of these techniques. While the three employed approaches demonstrate strong agreement, our analysis reveals that the EFDM results are in the best agreement with the analytical solutions. Given the consistent agreement between the numerical results from the EFDM, unconditionally positivity preserving FDM and PINN approach and the analytical solutions, all three methods are recommended as competitive options. The solution techniques employed in this study can be a valuable asset for present and future model developers engaged in various nonlinear physical wave phenomena, such as propagation of solitons in optical fibers. |
| format | Article |
| id | doaj-art-49607f2b4cb5441c8db4f6dca0b168a2 |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-49607f2b4cb5441c8db4f6dca0b168a22025-08-20T02:43:28ZengMDPI AGAxioms2075-16802024-12-01131287210.3390/axioms13120872Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference MethodsSvetislav Savović0Miloš Ivanović1Branko Drljača2Ana Simović3Faculty of Science, University of Kragujevac, 34000 Kragujevac, SerbiaFaculty of Science, University of Kragujevac, 34000 Kragujevac, SerbiaFaculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, 38220 Kosovska Mitrovica, SerbiaFaculty of Science, University of Kragujevac, 34000 Kragujevac, SerbiaThis study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated to demonstrate the effectiveness of these techniques. While the three employed approaches demonstrate strong agreement, our analysis reveals that the EFDM results are in the best agreement with the analytical solutions. Given the consistent agreement between the numerical results from the EFDM, unconditionally positivity preserving FDM and PINN approach and the analytical solutions, all three methods are recommended as competitive options. The solution techniques employed in this study can be a valuable asset for present and future model developers engaged in various nonlinear physical wave phenomena, such as propagation of solitons in optical fibers.https://www.mdpi.com/2075-1680/13/12/872Sine–Gordon equationnonlinear wave phenomenasolitonsfinite difference methodphysics-informed neural networks |
| spellingShingle | Svetislav Savović Miloš Ivanović Branko Drljača Ana Simović Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods Axioms Sine–Gordon equation nonlinear wave phenomena solitons finite difference method physics-informed neural networks |
| title | Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods |
| title_full | Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods |
| title_fullStr | Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods |
| title_full_unstemmed | Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods |
| title_short | Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods |
| title_sort | numerical solution of the sine gordon equation by novel physics informed neural networks and two different finite difference methods |
| topic | Sine–Gordon equation nonlinear wave phenomena solitons finite difference method physics-informed neural networks |
| url | https://www.mdpi.com/2075-1680/13/12/872 |
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