[Compensation du biais induit par le bruit pour l’apprentissage non supervisé des lois de comportement]
The Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) framework allows the non-supervised learning of constitutive laws from full-field displacement data and global reaction forces. Nonetheless, its accuracy is adversely affected by measurement noise, resulting in biased material parameter identification due to uniform nodal weighting and mesh dependencies. To mitigate these issues, covariance-based weighting and systematic bias compensation strategies are proposed, which account for measurement uncertainties and counteract noise-induced errors. Additionally, Gaussian smoothing is introduced as a low-cost alternative to reduce noise impact by averaging nodal force residuals. These methods were evaluated using both a simplified toy model and a complex numerical test case with realistic noise levels. Results demonstrate that the proposed compensation strategies significantly enhance EUCLID’s robustness and accuracy, achieving up to 93% improvement in validation metrics under high-noise conditions. Furthermore, mesh dependency issues are addressed, enabling mesh-independent learning. These advancements substantially improve the reliability of constitutive law identification in noisy experimental environments.
Le cadre EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery) permet l’apprentissage non supervisé de lois de comportement à partir de champs de déplacements et de forces globales. Toutefois, son exactitude est fortement dégradée en présence de bruit de mesure, ce qui induit une identification biaisée des paramètres matériaux en raison d’un poids nodal uniforme et d’une dépendance au maillage. Pour remédier à ces limitations, nous proposons une pondération fondée sur les covariances ainsi que des stratégies de compensation du biais, capables de tenir compte des incertitudes de mesure et de corriger les erreurs induites par le bruit. En complément, un lissage gaussien est introduit comme solution peu coûteuse pour réduire l’impact du bruit en moyennant les résidus nodaux de forces. Ces approches sont évaluées à la fois sur un modèle simplifié et sur un cas test numérique complexe intégrant des niveaux de bruit réalistes. Les résultats montrent que les stratégies de compensation proposées améliorent significativement la robustesse et la précision d’EUCLID, avec jusqu’à 93% d’amélioration des métriques de validation en conditions fortement bruitées. Par ailleurs, les problèmes de dépendance au maillage sont corrigés, permettant un apprentissage indépendant de la discrétisation. Ces avancées renforcent considérablement la fiabilité de l’identification de lois de comportement dans des environnements expérimentaux bruités.
Révisé le :
Accepté le :
Publié le :
Mots-clés : Intelligence artificielle, réseaux de neurones augmentés par la physique (PANN), EUCLID, bruit expérimental, corrélation d’images numériques, lois de comportement
Clément Jailin 1 ; Stéphane Roux 1 ; Antoine Benady 1 , 2 ; Emmanuel Baranger 1
CC-BY 4.0
@article{CRMECA_2026__354_G1_1_0,
author = {Cl\'ement Jailin and St\'ephane Roux and Antoine Benady and Emmanuel Baranger},
title = {Noise-bias compensation for the unsupervised learning of constitutive laws},
journal = {Comptes Rendus. M\'ecanique},
pages = {1--24},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {354},
doi = {10.5802/crmeca.342},
language = {en},
}
TY - JOUR AU - Clément Jailin AU - Stéphane Roux AU - Antoine Benady AU - Emmanuel Baranger TI - Noise-bias compensation for the unsupervised learning of constitutive laws JO - Comptes Rendus. Mécanique PY - 2026 SP - 1 EP - 24 VL - 354 PB - Académie des sciences, Paris DO - 10.5802/crmeca.342 LA - en ID - CRMECA_2026__354_G1_1_0 ER -
%0 Journal Article %A Clément Jailin %A Stéphane Roux %A Antoine Benady %A Emmanuel Baranger %T Noise-bias compensation for the unsupervised learning of constitutive laws %J Comptes Rendus. Mécanique %D 2026 %P 1-24 %V 354 %I Académie des sciences, Paris %R 10.5802/crmeca.342 %G en %F CRMECA_2026__354_G1_1_0
Clément Jailin; Stéphane Roux; Antoine Benady; Emmanuel Baranger. Noise-bias compensation for the unsupervised learning of constitutive laws. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 1-24. doi: 10.5802/crmeca.342
[1] A review on data-driven constitutive laws for solids, Arch. Comput. Methods Eng., Volume 32 (2025), pp. 1841-1883 | DOI
[2] Neural networks for constitutive modeling: from universal function approximators to advanced models and the integration of physics, Arch. Comput. Methods Eng., Volume 31 (2024) no. 2, pp. 1097-1127 | DOI
[3] Deep learning in computational mechanics: a review, Comput. Mech., Volume 74 (2024), pp. 281-331 | DOI | Zbl | MR
[4] Recent advances and applications of machine learning in experimental solid mechanics: a review, Appl. Mech. Rev., Volume 75 (2023) no. 6, 061001, 24 pages | DOI
[5] Data-based derivation of material response, Comput. Methods Appl. Mech. Eng., Volume 331 (2018), pp. 184-196 | DOI | Zbl | MR
[6] Non-parametric material state field extraction from full field measurements, Comput. Mech., Volume 64 (2019) no. 2, pp. 501-509 | DOI | Zbl | MR
[7] Model-free data-driven methods in mechanics: material data identification and solvers, Comput. Mech., Volume 64 (2019) no. 2, pp. 381-393 | DOI | Zbl | MR
[8] Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., Volume 378 (2019), pp. 686-707 | DOI | Zbl | MR
[9] Enhanced physics-informed neural networks for hyperelasticity, Int. J. Numer. Methods Eng., Volume 124 (2023) no. 7, pp. 1585-1601 | DOI | Zbl | MR
[10] Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN), Comput. Methods Appl. Mech. Eng., Volume 398 (2022), 115190 | DOI | Zbl | MR
[11] Neural networks meet hyperelasticity: a guide to enforcing physics, J. Mech. Phys. Solids, Volume 179 (2023), 105363 | DOI | MR
[12] Anisotropic hyperelastic constitutive models for finite deformations combining material theory and data-driven approaches with application to cubic lattice metamaterials, Comput. Mech., Volume 67 (2021), pp. 653-677 | DOI | Zbl | MR
[13] FE ANN: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining, Comput. Mech., Volume 71 (2023) no. 5, pp. 827-851 | DOI | MR | Zbl
[14] Input convex neural networks, Proceedings of the 34th International Conference on Machine Learning (Doina Precup; Yee Whye Teh, eds.) (Proceedings of Machine Learning Research), Volume 70, PMLR (2017), pp. 146-155
[15] Structure-preserving neural networks, J. Comput. Phys., Volume 426 (2021), 109950 | DOI | Zbl | MR
[16] Polyconvex anisotropic hyperelasticity with neural networks, J. Mech. Phys. Solids, Volume 159 (2022), 104703 | DOI | MR | Zbl
[17] Viscoelasticty with physics-augmented neural networks: model formulation and training methods without prescribed internal variables, Comput. Mech., Volume 74 (2024), pp. 1279-1301 | MR | DOI | Zbl
[18] Modular machine learning-based elastoplasticity: generalization in the context of limited data, Comput. Methods Appl. Mech. Eng., Volume 407 (2023), 115930 | DOI | Zbl | MR
[19] Benchmarking physics-informed frameworks for data-driven hyperelasticity, Comput. Mech., Volume 73 (2024), pp. 49-65 | Zbl | MR
[20] A comparative study on different neural network architectures to model inelasticity, Int. J. Numer. Methods Eng., Volume 124 (2023) no. 21, pp. 4802-4840 | DOI | Zbl | MR
[21] An iterative image registration technique with an application to stereo vision, IJCAI’81: Proceedings of the 7th International Joint Conference on Artificial Intelligence, Volume 2, Morgan-Kaufmann (1981), pp. 674-679
[22] Determination of displacements using an improved digital correlation method, Image Vision Comput., Volume 1 (1983) no. 3, pp. 133-139 | DOI
[23] On the future of experimental mechanics in the digital world: an eikological perspective, Eur. J. Mech. A Solids, Volume 113 (2025), 105654 | DOI | Zbl
[24] Methods and applications of digital volume correlation, J. Strain Anal. Eng. Des., Volume 43 (2008) no. 8, pp. 745-760
[25] Digital volume correlation: review of progress and challenges, Exp. Mech., Volume 58 (2018), pp. 661-708 | DOI
[26] Comparison of local and global approaches to digital image correlation, Exp. Mech., Volume 52 (2012) no. 9, pp. 1503-1519 | DOI
[27] Comprehensive full-field measurements via Digital Image Correlation, Compr. Mech. Mater., Volume 2 (2024), pp. 3-56 | DOI
[28] Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC, Exp. Mech., Volume 55 (2015), pp. 105-119 | DOI
[29] Optimal procedure for the identification of constitutive parameters from experimentally measured displacement fields, Int. J. Solids Struct., Volume 184 (2020), pp. 14-23 | DOI
[30] Unsupervised discovery of interpretable hyperelastic constitutive laws, Comput. Methods Appl. Mech. Eng., Volume 381 (2021), 113852, 19 pages | Zbl | MR
[31] Equilibrium-based convolution neural networks for constitutive modeling of hyperelastic materials, J. Mech. Phys. Solids, Volume 164 (2022), 104931 | DOI
[32] Experimental learning of a hyperelastic behavior with a physics-augmented neural network, Exp. Mech., Volume 64 (2024), pp. 1465-1481 | DOI
[33] Learning a hyperelastic constitutive model from 3D experimental data, Comput. Methods Appl. Mech. Eng., Volume 450 (2026), 118592, 22 pages | DOI | MR
[34] Physics-augmented neural networks for constitutive modeling, Ph. D. Thesis, Université Paris-Saclay (France) (2024)
[35] Material-embedding Physics-Augmented Neural Networks: a first application to constitutive law parameterization, Comput. Methods Appl. Mech. Eng., Volume 445 (2025), 118188, 18 pages | MR
[36] Unsupervised learning of history-dependent constitutive material laws with thermodynamically-consistent neural networks in the modified Constitutive Relation Error framework, Comput. Methods Appl. Mech. Eng., Volume 425 (2024), 116967 | DOI | MR
[37] NN-mCRE: a modified constitutive relation error framework for unsupervised learning of nonlinear state laws with physics-augmented neural networks, Int. J. Numer. Methods Eng., Volume 125 (2024) no. 8, e7439 | DOI | MR
[38] Data-driven finite elasticity, Arch. Ration. Mech. Anal., Volume 237 (2020) no. 1, pp. 1-33 | DOI | MR
[39] Finite element solver for data-driven finite strain elasticity, Comput. Methods Appl. Mech. Eng., Volume 379 (2021), 113756 | DOI | MR
[40] NN-EUCLID: Deep-learning hyperelasticity without stress data, J. Mech. Phys. Solids, Volume 169 (2022), 105076, 29 pages | DOI | MR
[41] The FEniCS project version 1.5, Arch. Numer. Soft., Volume 3 (2015) no. 100, pp. 9-23 | DOI
[42] A mechanics-informed artificial neural network approach in data-driven constitutive modeling, Int. J. Numer. Methods Eng., Volume 123 (2022) no. 12, pp. 2738-2759 | MR
[43] Adam: a method for stochastic optimization (2014) | arXiv
[44] The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements, Springer, 2012 | DOI
[45] Bayesian-EUCLID: Discovering hyperelastic material laws with uncertainties, Comput. Methods Appl. Mech. Eng., Volume 398 (2022), 115225, 36 pages | DOI | MR
[46] On the stress distribution at the base of a stationary crack, J. Appl. Mech., Volume 24 (1957) no. 1, pp. 109-114 | DOI | MR
[47] An extended and integrated digital image correlation technique applied to the analysis of fractured samples: the equilibrium gap method as a mechanical filter, Eur. J. Comput. Mech., Volume 18 (2009) no. 3–4, pp. 285-306 | DOI
Cité par Sources :
Commentaires - Politique