{"title":"Deep NURBS—admissible physics-informed neural networks","authors":"Hamed Saidaoui, Luis Espath, Raúl Tempone","doi":"10.1007/s00366-024-02040-9","DOIUrl":null,"url":null,"abstract":"<p>In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solutions for partial differential equations (PDEs) in case of arbitrary geometries while strongly enforcing Dirichlet boundary conditions. The proposed approach combines admissible NURBS parametrizations (admissible in the calculus of variations sense, that is, satisfying the boundary conditions) required to define the physical domain and the Dirichlet boundary conditions with a PINN solver. Therefore, the boundary conditions are automatically satisfied in this novel Deep NURBS framework. Furthermore, our sampling is carried out in the parametric space and mapped to the physical domain. This parametric sampling works as an importance sampling scheme since there is a concentration of points in regions where the geometry is more complex. We verified our new approach using two-dimensional elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the classical PINN solver, the Deep NURBS estimator has a remarkably high accuracy for all the studied problems. Moreover, a desirable accuracy was obtained for most of the studied PDEs using only one hidden layer of neural networks. This novel approach is considered to pave the way for more effective solutions for high-dimensional problems by allowing for a more realistic physics-informed statistical learning framework to solve PDEs.\n</p>","PeriodicalId":11696,"journal":{"name":"Engineering with Computers","volume":"41 1","pages":""},"PeriodicalIF":8.7000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering with Computers","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00366-024-02040-9","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solutions for partial differential equations (PDEs) in case of arbitrary geometries while strongly enforcing Dirichlet boundary conditions. The proposed approach combines admissible NURBS parametrizations (admissible in the calculus of variations sense, that is, satisfying the boundary conditions) required to define the physical domain and the Dirichlet boundary conditions with a PINN solver. Therefore, the boundary conditions are automatically satisfied in this novel Deep NURBS framework. Furthermore, our sampling is carried out in the parametric space and mapped to the physical domain. This parametric sampling works as an importance sampling scheme since there is a concentration of points in regions where the geometry is more complex. We verified our new approach using two-dimensional elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the classical PINN solver, the Deep NURBS estimator has a remarkably high accuracy for all the studied problems. Moreover, a desirable accuracy was obtained for most of the studied PDEs using only one hidden layer of neural networks. This novel approach is considered to pave the way for more effective solutions for high-dimensional problems by allowing for a more realistic physics-informed statistical learning framework to solve PDEs.
期刊介绍:
Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.