dc.contributor.author | Uriarte Baranda, Carlos | |
dc.contributor.author | Pardo Zubiaur, David | |
dc.contributor.author | Muga, Ignacio | |
dc.contributor.author | Muñoz Matute, Judit | |
dc.date.accessioned | 2023-06-19T17:42:07Z | |
dc.date.available | 2023-06-19T17:42:07Z | |
dc.date.issued | 2023-02 | |
dc.identifier.citation | Computer Methods in Applied Mechanics and Engineering 405 : (2023) // Article ID 115892 | es_ES |
dc.identifier.issn | 1879-2138 | |
dc.identifier.issn | 0045-7825 | |
dc.identifier.uri | http://hdl.handle.net/10810/61476 | |
dc.description.abstract | Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min–max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min–max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D2RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers. | 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 Marie Sklodowska-Curie individual fellowship No. 101017984 (GEODPG); the Spanish Ministry of Science and Innovation projects with references TED2021-132783B-I00, PID2019-108111RB-I00 (FEDER/AEI) and PDC2021-121093-I00 (AEI/Next Generation EU), the “BCAM Severo Ochoa” accreditation of excellence CEX2021-001142-S/MICIN/AEI/10.13039/501100011033; and the Basque Government through the BERC 2022–2025 program, the three Elkartek projects 3KIA (KK-2020/00049), EXPERTIA (KK-2021/00048), and SIGZE (KK-2021/00095), and the Consolidated Research Group MATHMODE (IT1456-22) given by the Department of Education | 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/EC/H2020/101017984 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/TED2021-132783B-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/PID2019-108111RB-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/PDC2021-121093-I00 | es_ES |
dc.relation | info:eu-repo/grantAgreement/MICINN/CEX2021-001142-S | 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 | partial differential equations | es_ES |
dc.subject | variational formulation | es_ES |
dc.subject | residual minimization | es_ES |
dc.subject | Ritz method | es_ES |
dc.subject | optimal test functions | es_ES |
dc.subject | neural networks | es_ES |
dc.title | A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks | es_ES |
dc.type | info:eu-repo/semantics/article | es_ES |
dc.rights.holder | © 2023 The Author(s). 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/S0045782523000154 | es_ES |
dc.identifier.doi | 10.1016/j.cma.2023.115892 | |
dc.contributor.funder | European Commission | |
dc.departamentoes | Matemáticas | es_ES |
dc.departamentoeu | Matematika | es_ES |