We propose and study a neural operator framework for learning memory- and material microstructure-dependent constitutive laws for heterogeneous materials. We work in the two-scale setting, where homogenization theory provides a systematic approach to deriving macroscale constitutive laws, obviating the need to resolve the complex microstructure repeatedly. However, the unit-cell problems that define these constitutive models are typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Our proposed framework models homogenized constitutive laws with both memory- and microstructure-dependence using Markovian recurrent and Fourier neural operators. The homogenization problem for Kelvin–Voigt viscoelastic materials is studied to provide firm theoretical foundations for our model. The theoretical properties of the cell problem in this Kelvin–Voigt setting motivate the proposed learning framework, and are also used to prove a universal approximation theorem for the learned macroscale constitutive model. Numerical experiments show that the proposed learning framework accurately learns memory- and microstructure-dependent viscoelastic and elasto-viscoplastic constitutive models, beyond the setting of the theory. Furthermore, we show that the learned constitutive models can be successfully deployed in macroscale simulations of material deformation for different microstructures without retraining.
Keywords: Constitutive modeling, homogenization, memory, microstructure, Kelvin–Voigt, viscoelasticity, elasto-viscoplasticity, neural operator
Kaushik Bhattacharya 1 ; Lianghao Cao 2 ; George Stepaniants 2 ; Andrew M. Stuart 2 ; Margaret Trautner 2
Kaushik Bhattacharya; Lianghao Cao; George Stepaniants; Andrew M. Stuart; Margaret Trautner. Learning Memory And Material Dependent Constitutive Laws. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 219-267. doi: 10.5802/smai-jcm.148
@article{SMAI-JCM_2026__12__219_0,
author = {Kaushik Bhattacharya and Lianghao Cao and George Stepaniants and Andrew M. Stuart and Margaret Trautner},
title = {Learning {Memory} {And} {Material} {Dependent} {Constitutive} {Laws}},
journal = {The SMAI Journal of computational mathematics},
pages = {219--267},
year = {2026},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
doi = {10.5802/smai-jcm.148},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.148/}
}
TY - JOUR AU - Kaushik Bhattacharya AU - Lianghao Cao AU - George Stepaniants AU - Andrew M. Stuart AU - Margaret Trautner TI - Learning Memory And Material Dependent Constitutive Laws JO - The SMAI Journal of computational mathematics PY - 2026 SP - 219 EP - 267 VL - 12 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.148/ DO - 10.5802/smai-jcm.148 LA - en ID - SMAI-JCM_2026__12__219_0 ER -
%0 Journal Article %A Kaushik Bhattacharya %A Lianghao Cao %A George Stepaniants %A Andrew M. Stuart %A Margaret Trautner %T Learning Memory And Material Dependent Constitutive Laws %J The SMAI Journal of computational mathematics %D 2026 %P 219-267 %V 12 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.148/ %R 10.5802/smai-jcm.148 %G en %F SMAI-JCM_2026__12__219_0
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