164041 20251121161351.0 doi 10.1016/j.cma.2025.118476 sideral 146303 ART-2025-146303 eng Tesán, Lucas Universidad de Zaragoza On the under-reaching phenomenon in message passing neural PDE solvers: Revisiting the CFL condition 2025 This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). This significantly reduces the need for exhaustive hyperparameter tuning. Bounds are derived for the three fundamental classes of PDEs (hyperbolic, parabolic and elliptic) by relating the physical characteristics of the problem in question to the message-passing requirement of GNNs. In particular, we investigate the relationship between the physical constants of the equations governing the problem, the spatial and temporal discretisation and the message passing mechanisms in GNNs. When the number of message passing iterations is below these proposed limits, information does not propagate efficiently through the network, resulting in poor solutions, even for deep GNN architectures. In contrast, when the suggested lower bound is satisfied, the GNN parameterisation allows the model to accurately capture the underlying phenomenology, resulting in solvers of adequate accuracy. Examples are provided for four different examples of equations that show the sharpness of the proposed lower bounds. Access copy available to the general public Unrestricted info:eu-repo/grantAgreement/ES/MICINN/PID2023-147373OB-I00 info:eu-repo/grantAgreement/ES/MTFP/TSI-100930-2023-1 info:eu-repo/semantics/openAccess by-nc-nd https://creativecommons.org/licenses/by-nc-nd/4.0/deed.es info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Martínez Iparraguirre, Mikel Universidad de Zaragoza González, David Universidad de Zaragoza (orcid)0000-0003-3003-5856 Lopes De Sousa Martins, Pedro (orcid)0000-0001-9732-4498 Cueto, Elías Universidad de Zaragoza (orcid)0000-0003-1017-4381 5004 605 Universidad de Zaragoza Dpto. Ingeniería Mecánica Área Mec.Med.Cont. y Teor.Est. 449 (2025), 118476 [17 pp.] Comput. methods appl. mech. eng. Computer Methods in Applied Mechanics and Engineering 0045-7825 7153362 http://zaguan.unizar.es/record/164041/files/texto_completo.pdf Versión publicada 1948703 http://zaguan.unizar.es/record/164041/files/texto_completo.jpg?subformat=icon icon Versión publicada oai:zaguan.unizar.es:164041 articulos driver 2025-11-21-14:25:20 ARTICLE