dc.contributor.author | Rivera González, Jon Ander | |
dc.contributor.author | Taylor, Jamie M. | |
dc.contributor.author | Omella Milian, Ángel Javier | |
dc.contributor.author | Pardo Zubiaur, David ![ORCID](/themes/Mirage2//images/orcid_16x16.png) | |
dc.date.accessioned | 2024-05-22T14:41:46Z | |
dc.date.available | 2024-05-22T14:41:46Z | |
dc.date.issued | 2022-04 | |
dc.identifier.citation | Computer Methods in Applied Mechanics and Engineering 393 : (2022) // Article ID 114710 | es_ES |
dc.identifier.issn | 1879-2138 | |
dc.identifier.issn | 0045-7825 | |
dc.identifier.uri | http://hdl.handle.net/10810/68100 | |
dc.description.abstract | Neural 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. | es_ES |
dc.description.sponsorship | This work has received funding from: the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 777778 (MATHROCKS); the European Regional Development Fund (ERDF) through the Interreg V-A Spain-France-Andorra program POCTEFA 2014–2020 Project PIXIL (EFA362/19); the Spanish Ministry of Science and Innovation projects with references PID2019-108111RB-I00 (FEDER/AEI), PDC 2021-121093-I00, and PID2020-114189RB-I00 and the “BCAM Severo Ochoa” accrediation of excellence (SEV-2017-0718); and the Basque Government, Spain through the three Elkartek projects 3KIA (KK-2020/00049), EXPERTIA (KK-2021/000 48), and SIGZE (KK-2021/00095), the Consolidated Research Group MATHMODE (IT1294-19) given by the Department of Education, and the BERC 2022–2025 program. | es_ES |
dc.language.iso | eng | es_ES |
dc.publisher | Elsevier | es_ES |
dc.relation | info:eu-repo/grantAgreement/EC/H2020/777778 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/PID2019-108111RB-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/PDC 2021-121093-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/PID2020-114189RB-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MINECO/SEV-2017-0718 | es_ES |
dc.rights | info:eu-repo/semantics/openAccess | es_ES |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/es/ | * |
dc.subject | deep learning | es_ES |
dc.subject | neural networks | es_ES |
dc.subject | Ritz method | es_ES |
dc.subject | least-squares method | es_ES |
dc.subject | quadrature rules | es_ES |
dc.title | On quadrature rules for solving Partial Differential Equations using Neural Networks | es_ES |
dc.type | info:eu-repo/semantics/article | es_ES |
dc.rights.holder | /© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:
//creativecommons.org/licenses/by-nc-nd/4.0/). | es_ES |
dc.rights.holder | Atribución-NoComercial-SinDerivadas 3.0 España | * |
dc.relation.publisherversion | https://www.sciencedirect.com/science/article/pii/S0045782522000810 | es_ES |
dc.identifier.doi | 10.1016/j.cma.2022.114710 | |
dc.contributor.funder | European Commission | |
dc.departamentoes | Matemática aplicada | es_ES |
dc.departamentoes | Matemáticas | es_ES |
dc.departamentoeu | Matematika | es_ES |
dc.departamentoeu | Matematika aplikatua | es_ES |