{"title":"An efficient wavelet-based physics-informed neural networks for singularly perturbed problems","authors":"Himanshu Pandey, Anshima Singh, Ratikanta Behera","doi":"arxiv-2409.11847","DOIUrl":null,"url":null,"abstract":"Physics-informed neural networks (PINNs) are a class of deep learning models\nthat utilize physics as differential equations to address complex problems,\nincluding ones that may involve limited data availability. However, tackling\nsolutions of differential equations with oscillations or singular perturbations\nand shock-like structures becomes challenging for PINNs. Considering these\nchallenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to\nsolve singularly perturbed differential equations. Here, we represent the\nsolution in wavelet space using a family of smooth-compactly supported\nwavelets. This framework represents the solution of a differential equation\nwith significantly fewer degrees of freedom while still retaining in capturing,\nidentifying, and analyzing the local structure of complex physical phenomena.\nThe architecture allows the training process to search for a solution within\nwavelet space, making the process faster and more accurate. The proposed model\ndoes not rely on automatic differentiations for derivatives involved in\ndifferential equations and does not require any prior information regarding the\nbehavior of the solution, such as the location of abrupt features. Thus,\nthrough a strategic fusion of wavelets with PINNs, W-PINNs excel at capturing\nlocalized nonlinear information, making them well-suited for problems showing\nabrupt behavior in certain regions, such as singularly perturbed problems. The\nefficiency and accuracy of the proposed neural network model are demonstrated\nin various test problems, i.e., highly singularly perturbed nonlinear\ndifferential equations, the FitzHugh-Nagumo (FHN), and Predator-prey\ninteraction models. The proposed design model exhibits impressive comparisons\nwith traditional PINNs and the recently developed wavelet-based PINNs, which\nuse wavelets as an activation function for solving nonlinear differential\nequations.","PeriodicalId":501301,"journal":{"name":"arXiv - CS - Machine Learning","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Machine Learning","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11847","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
Physics-informed neural networks (PINNs) are a class of deep learning models
that utilize physics as differential equations to address complex problems,
including ones that may involve limited data availability. However, tackling
solutions of differential equations with oscillations or singular perturbations
and shock-like structures becomes challenging for PINNs. Considering these
challenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to
solve singularly perturbed differential equations. Here, we represent the
solution in wavelet space using a family of smooth-compactly supported
wavelets. This framework represents the solution of a differential equation
with significantly fewer degrees of freedom while still retaining in capturing,
identifying, and analyzing the local structure of complex physical phenomena.
The architecture allows the training process to search for a solution within
wavelet space, making the process faster and more accurate. The proposed model
does not rely on automatic differentiations for derivatives involved in
differential equations and does not require any prior information regarding the
behavior of the solution, such as the location of abrupt features. Thus,
through a strategic fusion of wavelets with PINNs, W-PINNs excel at capturing
localized nonlinear information, making them well-suited for problems showing
abrupt behavior in certain regions, such as singularly perturbed problems. The
efficiency and accuracy of the proposed neural network model are demonstrated
in various test problems, i.e., highly singularly perturbed nonlinear
differential equations, the FitzHugh-Nagumo (FHN), and Predator-prey
interaction models. The proposed design model exhibits impressive comparisons
with traditional PINNs and the recently developed wavelet-based PINNs, which
use wavelets as an activation function for solving nonlinear differential
equations.