An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains
The simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary...
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2025-03-01
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| author | Laurence Finch Weizhong Dai Aniruddha Bora |
| author_facet | Laurence Finch Weizhong Dai Aniruddha Bora |
| author_sort | Laurence Finch |
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| description | The simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary to minimize the reflections of outgoing waves at the boundary, preventing interference with the simulation’s accuracy. However, the presence of higher-order partial derivatives, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub></semantics></math></inline-formula> in the Rosenau-KdV-RLW equation, raises challenges in deriving accurate artificial boundary conditions. To address this issue, we propose an artificial neural network (ANN) method that enables soliton propagation through the computational domain without imposing artificial boundary conditions. This method randomly selects training points from the bounded computational space-time domain, and the loss function is designed based solely on the initial conditions and the Rosenau-KdV-RLW equation itself, without any boundary conditions. We analyze the convergence of the ANN solution theoretically. This new ANN method is tested in three examples. The results indicate that the present ANN method effectively simulates soliton propagation based on the Rosenau-KdV-RLW equation in unbounded domains or over extended periods. |
| format | Article |
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| publishDate | 2025-03-01 |
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| spelling | doaj-art-8336eabfc0fd4a90816df111ed442d3a2025-08-20T03:08:57ZengMDPI AGMathematics2227-73902025-03-01137103610.3390/math13071036An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded DomainsLaurence Finch0Weizhong Dai1Aniruddha Bora2Program in Computational Analysis and Modeling, Louisiana Tech University, Ruston, LA 71272, USAMathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USADivision of Applied Mathematics, Brown University, Providence, RI 02906, USAThe simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary to minimize the reflections of outgoing waves at the boundary, preventing interference with the simulation’s accuracy. However, the presence of higher-order partial derivatives, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>u</mi><mrow><mi>x</mi><mi>x</mi><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub></semantics></math></inline-formula> in the Rosenau-KdV-RLW equation, raises challenges in deriving accurate artificial boundary conditions. To address this issue, we propose an artificial neural network (ANN) method that enables soliton propagation through the computational domain without imposing artificial boundary conditions. This method randomly selects training points from the bounded computational space-time domain, and the loss function is designed based solely on the initial conditions and the Rosenau-KdV-RLW equation itself, without any boundary conditions. We analyze the convergence of the ANN solution theoretically. This new ANN method is tested in three examples. The results indicate that the present ANN method effectively simulates soliton propagation based on the Rosenau-KdV-RLW equation in unbounded domains or over extended periods.https://www.mdpi.com/2227-7390/13/7/1036Rosenau-KdV-RLWwave propagationartificial neural networkunbounded domain |
| spellingShingle | Laurence Finch Weizhong Dai Aniruddha Bora An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains Mathematics Rosenau-KdV-RLW wave propagation artificial neural network unbounded domain |
| title | An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains |
| title_full | An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains |
| title_fullStr | An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains |
| title_full_unstemmed | An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains |
| title_short | An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains |
| title_sort | artificial neural network method for simulating soliton propagation based on the rosenau kdv rlw equation on unbounded domains |
| topic | Rosenau-KdV-RLW wave propagation artificial neural network unbounded domain |
| url | https://www.mdpi.com/2227-7390/13/7/1036 |
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