Deep learning-based schemes for singularly perturbed convection-diffusion problems
ESAIM. Proceedings, Tome 73 (2023), pp. 48-67
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Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.
Affiliations des auteurs :
Adrien Beguinet 1 ; Virginie Ehrlacher 2 ; Roberta Flenghi 3 ; Maria Fuente 4 ; Olga Mula 5 ; Agustin Somacal 6
@article{EP_2023_73_a3,
author = {Adrien Beguinet and Virginie Ehrlacher and Roberta Flenghi and Maria Fuente and Olga Mula and Agustin Somacal},
title = {Deep learning-based schemes for singularly perturbed convection-diffusion problems},
journal = {ESAIM. Proceedings},
pages = {48--67},
year = {2023},
volume = {73},
doi = {10.1051/proc/202373048},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202373048/}
}
TY - JOUR AU - Adrien Beguinet AU - Virginie Ehrlacher AU - Roberta Flenghi AU - Maria Fuente AU - Olga Mula AU - Agustin Somacal TI - Deep learning-based schemes for singularly perturbed convection-diffusion problems JO - ESAIM. Proceedings PY - 2023 SP - 48 EP - 67 VL - 73 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/202373048/ DO - 10.1051/proc/202373048 LA - en ID - EP_2023_73_a3 ER -
%0 Journal Article %A Adrien Beguinet %A Virginie Ehrlacher %A Roberta Flenghi %A Maria Fuente %A Olga Mula %A Agustin Somacal %T Deep learning-based schemes for singularly perturbed convection-diffusion problems %J ESAIM. Proceedings %D 2023 %P 48-67 %V 73 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/202373048/ %R 10.1051/proc/202373048 %G en %F EP_2023_73_a3
Adrien Beguinet; Virginie Ehrlacher; Roberta Flenghi; Maria Fuente; Olga Mula; Agustin Somacal. Deep learning-based schemes for singularly perturbed convection-diffusion problems. ESAIM. Proceedings, Tome 73 (2023), pp. 48-67. doi: 10.1051/proc/202373048
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