The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks
In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the un...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-02-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2023309?viewType=HTML |
| _version_ | 1828001882046464000 |
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| author | Sung Woong Cho Sunwoo Hwang Hyung Ju Hwang |
| author_facet | Sung Woong Cho Sunwoo Hwang Hyung Ju Hwang |
| author_sort | Sung Woong Cho |
| collection | DOAJ |
| description | In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known. |
| first_indexed | 2024-04-10T06:35:34Z |
| format | Article |
| id | doaj.art-c9fa0c0049ff47dd921bdf10859c5023 |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-10T06:35:34Z |
| publishDate | 2023-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-c9fa0c0049ff47dd921bdf10859c50232023-03-01T01:27:52ZengAIMS PressMathematical Biosciences and Engineering1551-00182023-02-012047154717010.3934/mbe.2023309The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networksSung Woong Cho0Sunwoo Hwang1Hyung Ju Hwang2Department of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of KoreaDepartment of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of KoreaDepartment of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of KoreaIn this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.https://www.aimspress.com/article/doi/10.3934/mbe.2023309?viewType=HTMLpartial differential equationtraveling wave solutionforward and inverse problemsphysics-informed neural networksmonotonic neural networksconvergence analysis |
| spellingShingle | Sung Woong Cho Sunwoo Hwang Hyung Ju Hwang The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks Mathematical Biosciences and Engineering partial differential equation traveling wave solution forward and inverse problems physics-informed neural networks monotonic neural networks convergence analysis |
| title | The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks |
| title_full | The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks |
| title_fullStr | The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks |
| title_full_unstemmed | The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks |
| title_short | The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks |
| title_sort | monotone traveling wave solution of a bistable three species competition system via unconstrained neural networks |
| topic | partial differential equation traveling wave solution forward and inverse problems physics-informed neural networks monotonic neural networks convergence analysis |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2023309?viewType=HTML |
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