With view on the context of convex thermomechanics, we propose tools based on the concept of Bregman divergence, a notion introduced in the 1960s and used in learning and optimization as well. This study is motivated by the need of “discrepancy measures” between physically constrained fields that are used both in traditional algorithms, analysis methods and data driven modelling or applications as well.
We give also a characterization of symmetrical Bregman divergences through their generating functions which can only be quadratic forms. Some properties of the Bregman divergence and the introduced concept of Bregman Gap between couples of dual quantities are given, and some existing errors in thermomechanics are recovered. Exploiting the framework of Standard Generalized Materials, we give a set of generating functions for a large range of applications, including coupled multi-physics. Finally some results useful for computational geometry are detailed.
Dans le contexte de la thermomécanique convexe, nous proposons des outils basés sur le concept de divergence de Bregman, une notion introduite dans les années 1960 et utilisée dans d’autres domaines aussi bien en apprentissage qu’en optimisation. Cette étude est motivée par le besoin de « mesures de dissimilarité » entre des champs physiquement contraints, champs qui sont utilisés à la fois dans les algorithmes traditionnels, les méthodes d’analyse et les applications ou les modélisations basées sur les données.
Nous donnons également une caractérisation des divergences de Bregman symétriques à travers leur fonction génératrice qui ne peut être qu’une fonction quadratique. Certaines propriétés de la divergence de Bregman et du concept introduit d’écart de Bregman entre les couples de quantités duales sont données, et certaines erreurs existantes en thermomécanique sont retrouvées. En exploitant le cadre des matériaux standard généralisés, nous donnons un ensemble de fonctions génératrices pour une large gamme d’applications, y compris la multiphysique couplée. Enfin, certains résultats utiles pour la géométrie computationnelle sont détaillés.
Revised:
Accepted:
Published online:
Mot clés : Thermomécanique, Convexité, Divergences de Bregman, Mesures de dissimilarité, Traitement de données
Stéphane Andrieux 1
CC-BY 4.0
@article{CRMECA_2023__351_G1_59_0,
author = {St\'ephane Andrieux},
title = {Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics},
journal = {Comptes Rendus. M\'ecanique},
pages = {59--81},
publisher = {Acad\'emie des sciences, Paris},
volume = {351},
year = {2023},
doi = {10.5802/crmeca.164},
language = {en},
}
TY - JOUR AU - Stéphane Andrieux TI - Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics JO - Comptes Rendus. Mécanique PY - 2023 SP - 59 EP - 81 VL - 351 PB - Académie des sciences, Paris DO - 10.5802/crmeca.164 LA - en ID - CRMECA_2023__351_G1_59_0 ER -
Stéphane Andrieux. Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics. Comptes Rendus. Mécanique, Volume 351 (2023), pp. 59-81. doi : 10.5802/crmeca.164. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.164/
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