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# Unsupervised learning of constitutive models with neural networks

A. Benady, E. Baranger, L. Chamoin, Clément Jailin, L. Lorenzis

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A. Benady, E. Baranger, L. Chamoin, Clément Jailin, L. Lorenzis. Unsupervised learning of constitutive models with neural networks. Data-driven mechanics and physics of solids, May 2025, Göteborg, Sweden. 10.23967/data-driven.2025.013. hal-05338457

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https://hal.science/hal-05338457v1

Submitted on 25 Nov 2025

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# Unsupervised learning of constitutive models with neural networks

Antoine Benady<fnref n="1" />,*, Emmanuel Baranger<fnref n="2" />, Ludovic Chamoin<fnref n="2" />, Clément Jailin<fnref n="2" />, Laura de Lorenzis<fnref n="1" />

In recent years, significant efforts have been made to model material behavior laws using neural networks (NN) [1, 2]. One of the key challenges in this field is ensuring thermodynamic consistency, which can be done by relying on the Generalized Standard Material (GSM) framework [3]. In the GSM framework, the constitutive model is described by two positive convex scalar functions, the Helmholtz free-energy and the dissipation potential. A recent trend involves leveraging input-convex neural networks to represent these two thermodynamic potentials, thereby inherently satisfying thermodynamic principles [4, 5].

These networks can be trained using measurable data within unsupervised frameworks, such as NN-EUCLID (see [2] for the original article and [6] for experimental validation) or NN-mCRE [4, 7]. The aim of this talk is to present recent advances in unsupervised learning applied to the modeling of mechanical constitutive laws. In particular, the two previously mentioned unsupervised approaches will be compared with respect to criteria such as computational efficiency, noise robustness, type of data required, and sensitivity to initialization.

## References

[1] D.K. Klein, M. Fernández, R.J. Martin, P. Neff, O. Weeger, (2022). Polyconvex anisotropic hyperelasticity with neural networks. *Journal of the Mechanics and Physics of Solids*, 159, 104703.

[2] P. Thakolkaran, A. Joshi, Y. Zheng, M. Flaschel, L. De Lorenzis, S. Kumar, (2022). NN-EUCLID: Deep-learning hyperelasticity without stress data. *Journal of the Mechanics and Physics of Solids*, 169, 105076.

[3] B. Halphen, Q.S. Nguyen. Sur les matériaux standards généralisés. 1975

[4] A. Benady, E. Baranger, L. Chamoin, (2024). Unsupervised learning of history-dependent constitutive material laws with thermodynamically consistent neural networks in the modified Constitutive Relation Error framework. *Computer Methods in Applied Mechanics and Engineering*, 425, 116967.

[5] M. Rosenkranz, K.A. Kalina, J. Brummund, M. Kästner, (2024). A comparative study on different neural network architectures to model inelasticity. *Numerical Methods in Engineering*, 124, 4802–4840

[6] C. Jailin, A. Benady, R. Legroux, E. Baranger. (2024) Experimental Learning of a Hyperelastic Behavior with a Physics-Augmented Neural Network. *Experimental Mechanics*

[7] A. Benady, E. Baranger, L. Chamoin, (2024). 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.* 125(8):e7439.

<footnote n="1">¹ Department of Mechanical and Process Engineering, ETH Zürich, 8092 Zürich, Switzerland</footnote>

<footnote n="2">² Université Paris-Saclay, CentraleSupelec, ENS Paris-Saclay, CNRS, LMPS-Laboratoire de mécanique Paris-Saclay, Gif-sur-Yvette, France</footnote>

<footnote>*Corresponding author: abenady@ethz.ch</footnote>