Publicação
Error estimates and generalized trial constructions for solving ODEs using physics-informed neural networks
| authorProfile.email | biblioteca@isel.pt | |
| datacite.subject.fos | Ciências Naturais::Matemáticas | |
| dc.contributor.author | Babni, Atmane | |
| dc.contributor.author | Jamiai, Ismail | |
| dc.contributor.author | Rodrigues, José Alberto | |
| dc.date.accessioned | 2026-01-08T10:32:09Z | |
| dc.date.available | 2026-01-08T10:32:09Z | |
| dc.date.issued | 2025-11-24 | |
| dc.description.abstract | In this paper, we address the challenge of solving differential equations using physics-informed neural networks (PINNs), an innovative approach that integrates known physical laws into neural network training. The PINN approach involves three main steps: constructing a neural-network-based solution ansatz, defining a suitable loss function, and minimizing this loss via gradient-based optimization. We review two primary PINN formulations: the standard PINN I and an enhanced PINN II. The latter explicitly incorporates initial, final, or boundary conditions. Focusing on first-order differential equations, PINN II methods typically express the approximate solution as u˜(x,θ)=P(x)+Q(x)N(x,θ), where N(x,θ) is the neural network output with parameters θ, and P(x) and Q(x) are polynomial functions. We generalize this formulation by replacing the polynomial Q(x) with a more flexible function ϕ(x). We demonstrate that this generalized form yields a uniform approximation of the true solution, based on Cybenko’s universal approximation theorem. We further show that the approximation error diminishes as the loss function converges. Numerical experiments validate our theoretical findings and illustrate the advantages of the proposed choice of ϕ(x). Finally, we outline how this framework can be extended to higher-order or other classes of differential equations. | eng |
| dc.identifier.citation | Babni, A., Jamiai, I., & Rodrigues, J. A. (2025). Error estimates and generalized trial constructions for solving ODEs using physics-informed neural networks. Mathematical and Computational Applications, 30(6), 127. https://doi.org/10.3390/mca30060127 | |
| dc.identifier.doi | 10.3390/mca30060127 | |
| dc.identifier.eissn | 2297-8747 | |
| dc.identifier.uri | http://hdl.handle.net/10400.21/22454 | |
| dc.language.iso | eng | |
| dc.peerreviewed | yes | |
| dc.publisher | MDPI AG | |
| dc.relation.hasversion | https://www.mdpi.com/2297-8747/30/6/127 | |
| dc.relation.ispartof | Mathematical and Computational Applications | |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | |
| dc.subject | Physics-informed neural networks (PINNs) | |
| dc.subject | Differential equations | |
| dc.subject | Universal approximation theorem | |
| dc.subject | Loss function | |
| dc.subject | Error estimates | |
| dc.title | Error estimates and generalized trial constructions for solving ODEs using physics-informed neural networks | eng |
| dc.type | journal article | |
| dspace.entity.type | Publication | |
| oaire.citation.endPage | 26 | |
| oaire.citation.issue | 6 | |
| oaire.citation.startPage | 1 | |
| oaire.citation.title | Mathematical and Computational Applications | |
| oaire.citation.volume | 30 | |
| oaire.version | http://purl.org/coar/version/c_be7fb7dd8ff6fe43 | |
| person.familyName | Rodrigues | |
| person.givenName | José Alberto | |
| person.identifier.ciencia-id | 241B-90A4-D998 | |
| person.identifier.orcid | 0000-0001-5630-7149 | |
| person.identifier.scopus-author-id | 7202707426 | |
| relation.isAuthorOfPublication | 6743a818-d8d5-489f-829e-43391b3257cc | |
| relation.isAuthorOfPublication.latestForDiscovery | 6743a818-d8d5-489f-829e-43391b3257cc |