Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga
{"title":"Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations","authors":"Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga","doi":"arxiv-2409.08941","DOIUrl":null,"url":null,"abstract":"Reaction-Diffusion systems arise in diverse areas of science and engineering.\nDue to the peculiar characteristics of such equations, analytic solutions are\nusually not available and numerical methods are the main tools for\napproximating the solutions. In the last decade, artificial neural networks\nhave become an active area of development for solving partial differential\nequations. However, several challenges remain unresolved with these methods\nwhen applied to reaction-diffusion equations. In this work, we focus on two\nmain problems. The implementation of homogeneous Neumann boundary conditions\nand long-time integrations. For the homogeneous Neumann boundary conditions, we\nexplore four different neural network methods based on the PINN approach. For\nthe long time integration in Reaction-Diffusion systems, we propose a domain\nsplitting method in time and provide detailed comparisons between different\nimplementations of no-flux boundary conditions. We show that the domain\nsplitting method is crucial in the neural network approach, for long time\nintegration in Reaction-Diffusion systems. We demonstrate numerically that\ndomain splitting is essential for avoiding local minima, and the use of\ndifferent boundary conditions further enhances the splitting technique by\nimproving numerical approximations. To validate the proposed methods, we\nprovide numerical examples for the Diffusion, the Bistable and the Barkley\nequations and provide a detailed discussion and comparisons of the proposed\nmethods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"49 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08941","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Reaction-Diffusion systems arise in diverse areas of science and engineering.
Due to the peculiar characteristics of such equations, analytic solutions are
usually not available and numerical methods are the main tools for
approximating the solutions. In the last decade, artificial neural networks
have become an active area of development for solving partial differential
equations. However, several challenges remain unresolved with these methods
when applied to reaction-diffusion equations. In this work, we focus on two
main problems. The implementation of homogeneous Neumann boundary conditions
and long-time integrations. For the homogeneous Neumann boundary conditions, we
explore four different neural network methods based on the PINN approach. For
the long time integration in Reaction-Diffusion systems, we propose a domain
splitting method in time and provide detailed comparisons between different
implementations of no-flux boundary conditions. We show that the domain
splitting method is crucial in the neural network approach, for long time
integration in Reaction-Diffusion systems. We demonstrate numerically that
domain splitting is essential for avoiding local minima, and the use of
different boundary conditions further enhances the splitting technique by
improving numerical approximations. To validate the proposed methods, we
provide numerical examples for the Diffusion, the Bistable and the Barkley
equations and provide a detailed discussion and comparisons of the proposed
methods.