Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics

Stéphane Andrieux

doi:10.5802/crmeca.164

Note

Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics
[Mesures de dissimilarité par divergences de Bregman pour le calcul et l’apprentissage en thermomécanique]

Stéphane Andrieux ¹

¹ ONERA, Université Paris-Saclay, Chemin de la Hunière, 91123 Palaiseau, France

Comptes Rendus. Mécanique, Volume 351 (2023), pp. 59-81.

Résumé
Abstract

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.

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.

Reçu le : 2022-09-06
Révisé le : 2022-11-18
Accepté le : 2022-12-20
Publié le : 2023-02-01

DOI : 10.5802/crmeca.164

Keywords: Thermomechanics, Convexity, Bregman divergences, Discrepancy measures, Data processing
Mot clés : Thermomécanique, Convexité, Divergences de Bregman, Mesures de dissimilarité, Traitement de données

Affiliations des auteurs :

Stéphane Andrieux ¹

¹ ONERA, Université Paris-Saclay, Chemin de la Hunière, 91123 Palaiseau, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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  -

%0 Journal Article
%A Stéphane Andrieux
%T Bregman divergences for physically informed discrepancy measures for learning and computation in thermomechanics
%J Comptes Rendus. Mécanique
%D 2023
%P 59-81
%V 351
%I Académie des sciences, Paris
%R 10.5802/crmeca.164
%G en
%F CRMECA_2023__351_G1_59_0

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/

Bibliographie
Cité par

[1] C. C. Aggarwal; A. Hinneburg; D. A. Keim On the surprising behavior of distance metrics in high dimensional space, Database Theory — ICDT 2001 (J. Van den Bussche; V. Vianu, eds.) (Lecture Notes in Computer Science), Volume 1973, Springer, Berlin, Heidelberg, 2001, pp. 420-434 | DOI | Zbl

[2] K. Beyer; J. Goldstein; R. Ramakrishnan; U. Schaft When is “nearest neighbor” meaningful?, Database Theory—ICDT’99. ICDT 1999 (C. Beeri; P. Buneman, eds.) (Lecture Notes in Computer Science), Volume 1540, Springer, Berlin, Heidelberg, 1999 | DOI

[3] R. V. Kohn; M Vogelius Relaxation of a variational method for impedance computed tomography, Commun. Pure Appl. Math., Volume 40 (1987), pp. 745-777 | DOI | Zbl

[4] P. Ladevèze; D. Leguillon Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., Volume 20 (1983), pp. 485-509 | DOI | Zbl

[5] N. Moës; P. Ladevèze; B. Douchin Constitutive relation error estimators for (visco)plastic finite element analysis with softening, Comput. Methods Appl. Mech. Eng., Volume 176 (1999), pp. 247-264 | Zbl

[6] P. Feissel; O. Allix Modified constitutive relation error identification strategy for transient dynamics with corrupted data: the elastic case, Appl. Mech. Eng., Volume 196 (2007), pp. 1968-1983 | DOI | Zbl

[7] S. Andrieux; A. Ben Adba; H. D. Bui Sur identification de fissures planes via le concept d’écart à la réciprocité, C. R. Acad. Sci. Paris Sér. II, Volume 324 (1997), pp. 431-438 | Zbl

[8] D. Claire; F. Hild; S. Roux Identification of damage fields using kinematics measurements, C. R. Méc., Volume 330 (2002), pp. 729-734 | DOI | Zbl

[9] S. Andrieux; T. N. Baranger Solution of nonlinear Cauchy problem for hyperelastic solids, Inverse Probl., Volume 31 (2015) no. 11, pp. 115003-115022 | DOI | Zbl

[10] S. Andrieux; T. N. Baranger On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries, J. Mech. Phys. Solids, Volume 97 (2016), pp. 140-155 | DOI | Zbl

[11] S. Andrieux; T. N. Baranger Nonlinear Cauchy problem and identification in contact mechanics: a solving method based on Bregman-gap, Inverse Probl., Volume 36 (2020) no. 11, 115012 | DOI | Zbl

[12] L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., Volume 7 (1967) no. 3, pp. 200-217 | DOI | Zbl

[13] K. C. Kiwiel Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., Volume 35 (1997) no. 4, pp. 1142-1168 | DOI | Zbl

[14] Y. Censor; A. Lent An iterative row-action method for interval convex programming, J. Optim. Theory Appl., Volume 34 (1981) no. 3, pp. 321-353 | DOI | Zbl

[15] C. Basserville Divergence measures for statistical data processing–An annotated bibliography, Signal Process., Volume 93 (2013), pp. 621-633 | DOI

[16] A. Banerjee; S. Merugu; I. S. Dhillon; J. Ghosh Clustering with Bregman divergences, J. Mach. Learn. Res., Volume 6 (2005), pp. 1705-1749 | Zbl

[17] R.-A. Dragomir Bregman gradient methods for relatively-smooth optimization, Ph. D. Thesis, UT1 Capitole, France (2021)

[18] B. A. Frigyik; S. Srivastava; M. R. Gupta Functional Bregman divergence and Bayesian estimation of distributions, IEEE Trans. Inf. Theory, Volume 54 (2008) no. 11, pp. 1681-1685 | DOI | Zbl

[19] Z. Zhang; B. C. Ooi; S. Parthasarathy; A. K. H. Tung Similarity search on Bregman divergence: towards non-metric indexing, 35th International Conference on Very Large Data Bases, VLBD09, Lyon, France (2009)

[20] H. Jeffreys An invariant form of the prior probability in estimation problems, Proc. R. Soc. Lond., Volume 186 (1946), p. 463-461 | Zbl

[21] F. Nielsen; R. Nock Skew Jensen-Bregman Voronoi diagrams, Transactions on Computational Science XIV (Lecture Notes in Computer Science), Volume 6970, Springer, Berlin, Heidelberg, 2011, pp. 102-128 | DOI | Zbl

[22] P. Chen; Y. Chen; M. Rao Metrics defined by Bregman divergences, Commun. Math. Sci., Volume 6 (2008), pp. 915-926 | DOI | Zbl

[23] L. Wu; C. H. Choi; R. Jin; J. Zhu; N. Yu Learning Bregman distance for semi supervised clustering, IEEE Trans. Knowl. Data Eng., Volume 24 (2012) no. 3, pp. 478-491 | DOI

[24] R. T. Rockafellar Convex Analysis, Princeton University Press, Princeton, NJ, 1970 | DOI

[25] I. Ekeland; R. Témam Convex Analysis and Variational Problems, SIAM, Philadephia, 1999 | DOI

[26] J. B. Hiriart-Urruty; C. Lemarechal Convex analysis and minimization algorithms II, Advanced Theory and Bundle Methods (Grundlehren der mathematischen Wissenschaften), Volume 306, Springer, Berlin, Heidelberg, 2013

[27] A Brøndsted; R. T. Rockafellar On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., Volume 16 (1965), pp. 605-611 | DOI | Zbl

[28] S. Andrieux; T. N. Baranger; A. Ben Abda Solving Cauchy problem by minimizing an energy-like functional, Inverse Probl., Volume 22 (2006), pp. 115-133 | DOI | Zbl

[29] B. Halphen; Q. S. Nguyen Sur les matériaux standards généralisés, J. Méc., Volume 14 (1975), pp. 39-63 | Zbl

[30] D. C. Drucker A more fundamental approach to plastic stress-strain relations, Proceedings of the First US Congress of Applied Mechanics, ASME, New York (1951), pp. 487-491

[31] J. J. Marigo From Clausius-Duhem and Drücker-Ilyushin inequalities to standard materials, Continuum Thermomechanics: The Art and Science of Modelling Material behaviour, Paul Germain’s Anniversary Volume (A. Maugin; R. Douot; F. Sidoroff, eds.) (Solid Mechanics and Its Applications), Volume 76, Kluwer Academic Publishers, Springer, Dordrecht, 2000

[32] P. Germain; Q. S. Nguyen; P. Suquet Continuum thermodynamics, Trans. ASME, Volume 50 (1983), pp. 1010-1020 | DOI | Zbl

[33] J. M. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., Volume 65 (1977), pp. 193-201

[34] G. de Saxcé; Z. Q. Feng New inequation and functional for contact with friction: the implicit standard material approach, Mech. Struct. Mach., Volume 19 (1991) no. 3, pp. 301-325 | DOI

[35] A. Banerjee; X. Guo; H. Wang On the optimality condition expectations as a Bregman predictor, IEEE Trans. Inf. Theory, Volume 51 (2005), pp. 2664-2669 | DOI

[36] R. Nock; F. Nielsen Fitting the smallest enclosing Bregman ball, Machine Learning: ECML 2005 (J. Gama; R. Camacho; P. B. Brazdil; A. M. Jorge; L. Torgo, eds.) (Lecture Notes in Computer Science), Volume 3720, Springer, Berlin, 2005

[37] J. D. Boissonnat; F. Nielsen; R. Nock Bregman Voronoi diagrams, Discrete Comput. Geom., Volume 44 (2010), pp. 281-307 | DOI | Zbl

[38] F. Nielsen; R. Nock On the smallest enclosing information disk, Inf. Process. Lett., Volume 105 (2005), pp. 93-97 | DOI | Zbl

[39] A. Fischer Quantization and clustering with Bregman divergences, J. Multivar. Anal., Volume 101 (2010) no. 9, pp. 2207-2221 | DOI | Zbl

[40] M. Luci; O. Bachem; A. Krause String coresets for hard and soft Bregman Clustering with application to exponential family mixture, Proceedings of 19th International Conference on Artificial Intelligence and Statistics (AISTATS), Cadiz, Spain (2016)

[41] L. Cayton Efficient Bregman range search, Advances in Neural Information Processing Systems 22–Proceedings of the 2009 Conference (2009), pp. 243-251

[42] M. R. Ackermann; J. Blömer Coresets and approximate clustering for Bregman divergences, International Symposium on Discrete Algorithms (SODA), SIAM (2009), pp. 1088-1097 | Zbl

[43] L. Cayton Fast nearest neighbor retrieval for Bregman divergences, Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland (2008) | DOI

[44] P. G. Ciarlet Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity, North Holland, Amsterdam, 1988

Cité par Sources :

Commentaires - Politique