Non-smooth optimization meets automated material model discovery
- 1Department of Mechanical Engineering, Stanford University, 440 Escondido Mall, California, 94305, United States
- 2Institute of Applied Mechanics, Egerlandstraße 5, Friedrich-Alexander-Universität Erlangen–Nürnberg, Erlangen, 91058, Germany
- 3Department of Statistics, Stanford University, Sequoia Hall 390 Jane Stanford Way, California, 94305-4065, United States
Abstract
Automated material model discovery has gained significant traction in recent years, as it disrupts the tedious and time-consuming cycle of iteratively calibrating and modifying manually designed models. Non-smooth L 1 -norm regularization is the backbone of automated model discovery; however, the current literature on automated material model discovery offers limited insights into the robust and efficient minimization of non-smooth objective functions. In this work, we examine the minimization of functions of the form f ( w ) + α ∥ w ∥ 1 , where w are the material model parameters, f is a metric that quantifies the mismatch between the material model and the observed data, and α ≥ 0 is a regularization parameter that determines the sparsity of the solution. We investigate both the straightforward case where f is quadratic and the more complex scenario where it is non-quadratic or even non-convex. Importantly, in contrast to previous works on automated material model discovery, we do not only focus on methods that solve the sparse regression problem for a given value of the regularization parameter α , but propose methods to efficiently compute the entire regularization path, facilitating the selection of a suitable α . Specifically, we present four algorithms and discuss their roles for automated material model discovery in mechanics: First, we recapitulate a well-known coordinate descent algorithm that solves the minimization problem assuming that f is quadratic for a given value of α , also known as the LASSO. Second, we discuss the algorithm LARS, which automatically determines the critical values of α , at which material parameters in w are set to zero. Third, we propose to use the proximal gradient method ISTA for automated material model discovery if f is not quadratic, and fourth, we suggest a novel pathwise extension of ISTA for computing the regularization path. Many of these algorithms have not yet been applied to problems in computational material mechanics. We demonstrate the applicability of all algorithms for the automated discovery of incompressible hyperelastic material models from uniaxial tension and simple shear data.
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BibTeX
@article{flaschel2026nonsmooth,
title = {{Non-smooth optimization meets automated material model discovery}},
author = {Moritz Flaschel and Trevor J. Hastie and Ellen Kuhl A},
journal = {Journal of Computational Physics},
year = {2026},
month = may,
doi = {10.1016/j.jcp.2026.115043},
url = {https://doi.org/10.1016/j.jcp.2026.115043},
note = {CC BY},
}
RIS
TY - JOUR
TI - Non-smooth optimization meets automated material model discovery
AU - Moritz Flaschel
AU - Trevor J. Hastie
AU - Ellen Kuhl A
PY - 2026
DA - 2026/05/19
JO - Journal of Computational Physics
DO - 10.1016/j.jcp.2026.115043
UR - https://doi.org/10.1016/j.jcp.2026.115043
AB - Automated material model discovery has gained significant traction in recent years, as it disrupts the tedious and time-consuming cycle of iteratively calibrating and modifying manually designed models. Non-smooth L 1 -norm regularization is the backbone of automated model discovery; however, the current literature on automated material model discovery offers limited insights into the robust and efficient minimization of non-smooth objective functions. In this work, we examine the minimization of functions of the form f ( w ) + α ∥ w ∥ 1 , where w are the material model parameters, f is a metric that quantifies the mismatch between the material model and the observed data, and α ≥ 0 is a regularization parameter that determines the sparsity of the solution. We investigate both the straightforward case where f is quadratic and the more complex scenario where it is non-quadratic or even non-convex. Importantly, in contrast to previous works on automated material model discovery, we do not only focus on methods that solve the sparse regression problem for a given value of the regularization parameter α , but propose methods to efficiently compute the entire regularization path, facilitating the selection of a suitable α . Specifically, we present four algorithms and discuss their roles for automated material model discovery in mechanics: First, we recapitulate a well-known coordinate descent algorithm that solves the minimization problem assuming that f is quadratic for a given value of α , also known as the LASSO. Second, we discuss the algorithm LARS, which automatically determines the critical values of α , at which material parameters in w are set to zero. Third, we propose to use the proximal gradient method ISTA for automated material model discovery if f is not quadratic, and fourth, we suggest a novel pathwise extension of ISTA for computing the regularization path. Many of these algorithms have not yet been applied to problems in computational material mechanics. We demonstrate the applicability of all algorithms for the automated discovery of incompressible hyperelastic material models from uniaxial tension and simple shear data.
SN - 0021-9991
SN - 1090-2716
ER -