Show simple item record

dc.contributor.authorRivera, J.A 
dc.contributor.authorTaylor, J.M. 
dc.contributor.authorOmella, Ángel J.
dc.contributor.authorPardo, D. 
dc.date.accessioned2022-03-07T14:23:06Z
dc.date.available2022-03-07T14:23:06Z
dc.date.issued2022-04-01
dc.identifier.issn00457825
dc.identifier.urihttp://hdl.handle.net/20.500.11824/1440
dc.description.abstractNeural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose several alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed numerical integration scheme. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks.en_US
dc.formatapplication/pdfen_US
dc.language.isoengen_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/es/en_US
dc.subjectDeep learningen_US
dc.subjectLeast-Squares methoden_US
dc.subjectNeural Networksen_US
dc.subjectQuadrature rulesen_US
dc.subjectRitz methoden_US
dc.titleOn quadrature rules for solving Partial Differential Equations using Neural Networksen_US
dc.typeinfo:eu-repo/semantics/articleen_US
dc.identifier.doi10.1016/j.cma.2022.114710en_US
dc.relation.publisherversionhttps://www.sciencedirect.com/science/article/pii/S0045782522000810en_US
dc.relation.projectIDinfo:eu-repo/grantAgreement/EC/H2020/777778en_US
dc.relation.projectIDES/1PE/SEV-2017-0718en_US
dc.relation.projectIDES/2PE/PID2019-108111RB-I00en_US
dc.relation.projectIDES/2PE/PID2020-114189RB-I00en_US
dc.relation.projectIDEUS/BERC/BERC.2018-2021en_US
dc.relation.projectIDEUS/ELKARTEKen_US
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessen_US
dc.type.hasVersioninfo:eu-repo/semantics/publishedVersionen_US
dc.journal.titleComputer Methods in Applied Mechanics and Engineeringen_US
dc.volume.number393en_US


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record

Reconocimiento-NoComercial-CompartirIgual 3.0 España
Except where otherwise noted, this item's license is described as Reconocimiento-NoComercial-CompartirIgual 3.0 España