{"title":"Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation","authors":"Jiajing Guan, Howard Elman","doi":"arxiv-2409.07671","DOIUrl":null,"url":null,"abstract":"Singularly perturbed problems are known to have solutions with steep boundary\nlayers that are hard to resolve numerically. Traditional numerical methods,\nsuch as Finite Difference Methods (FDMs), require a refined mesh to obtain\nstable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have\nbeen shown to successfully approximate solutions to differential equations from\nvarious fields, it is natural to examine their performance on singularly\nperturbed problems. The convection-diffusion equation is a representative\nexample of such a class of problems, and we consider the use of PINNs to\nproduce numerical solutions of this equation. We study two ways to use PINNS:\nas a method for correcting oscillatory discrete solutions obtained using FDMs,\nand as a method for modifying reduced solutions of unperturbed problems. For\nboth methods, we also examine the use of input transformation to enhance\naccuracy, and we explain the behavior of input transformations analytically,\nwith the help of neural tangent kernels.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","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-2409.07671","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
Singularly perturbed problems are known to have solutions with steep boundary
layers that are hard to resolve numerically. Traditional numerical methods,
such as Finite Difference Methods (FDMs), require a refined mesh to obtain
stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have
been shown to successfully approximate solutions to differential equations from
various fields, it is natural to examine their performance on singularly
perturbed problems. The convection-diffusion equation is a representative
example of such a class of problems, and we consider the use of PINNs to
produce numerical solutions of this equation. We study two ways to use PINNS:
as a method for correcting oscillatory discrete solutions obtained using FDMs,
and as a method for modifying reduced solutions of unperturbed problems. For
both methods, we also examine the use of input transformation to enhance
accuracy, and we explain the behavior of input transformations analytically,
with the help of neural tangent kernels.