Benchmarking data-driven material models on the classic Treloar dataset

Hagen Holthusen; Moritz Flaschel; Denisa Martonová; Ellen Kuhl

cs.AIAug 14, 2026

Benchmarking data-driven material models on the classic Treloar dataset

Authors: Hagen Holthusen, Moritz Flaschel, Denisa Martonová, Ellen Kuhl

Organizations: Institute of Applied Mechanics, Egerlandstraße 5, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany · Department of Mechanical Engineering, Stanford University, 440 Escondido Mall, California 94305, United States.

Abstract

Machine learning is rapidly reshaping constitutive modeling, offers new ways to learn material behavior directly from experimental data, and challenges long-established modeling paradigms. But with a growing number of machine-learning-based approaches available, how do they compare in practice? In this paper, we use the classic experimental data of Treloar to benchmark popular frameworks for hyperelasticity: (Generalized-Invariant) Constitutive Artificial Neural Networks, Physics-Augmented Neural Networks, (Adaptive) Material Fingerprinting, and Efficient Unsupervised Constitutive Law Identification & Discovery. We compare their fitting performance, computational cost, hyperparameter sensitivity, and ease of implementation. Furthermore, we discuss the trade-offs between predictive accuracy and model complexity. The latter is assessed by quantifying both the number of material parameters in the discovered models and the computational time required to evaluate the constitutive model and its derivatives. The results show that all methods can reproduce the benchmark data remarkably well. Rather than identifying a single winner, we highlight the strengths and limitations of each approach and provide practical guidance for their use. The source code for all six methods, including the training and comparison scripts, as well as all results and data used in this study, is publicly available via https://doi.org/10.5281/zenodo.21915635.

Figures & tables

Figure 1: Strain energy density in the principal stretch space predicted by the six selected best-performing models (see Section 4 for details).

Figure 2: Comparison of Treloar’s experimental data and predictions from the best-performing discovered models.

Method	Solvers / Algorithms	Hyperparameters	Modeling Decisions
EUCLID	$\bullet$ Coordinate Descent (CD) $\bullet$ Least Angle Regression (LARS)	$\bullet$ Number of terms in the library $\bullet$ Regularization parameter $\alpha$ $\bullet$ Thresholding parameter $\theta_{{\mathrel{\scalebox{0.5}[1]{$ \rightarrow $}}}{0}}$	$\bullet$ Choice of the library features $\bullet$ Model input variables
Material Fingerprinting	$\bullet$ Pattern-recognition	$\bullet$ Number of fingerprints in the database $\bullet$ Parameter sampling during database generation	$\bullet$ Choice of the models in the database $\bullet$ Model input variables
Adaptive Material Fingerprinting	$\bullet$ Adaptive pattern-recognition	$\bullet$ Number of fingerprints in the database $\bullet$ Parameter sampling during database generation $\bullet$ Number of terms in the model $N_{a}$ $\bullet$ Step size $s\in[0,1]$	$\bullet$ Choice of the modeling features in the database $\bullet$ Model input variables
PANN	Various (usually gradient-based) $\bullet$ SGD $\bullet$ Adam $\bullet$ L-BFGS	$\bullet$ Learning rate $\eta$ $\bullet$ Number of epochs $\bullet$ Number of layers $\bullet$ Number of neurons per layer $\bullet$ Optional ML hyperparameters (e.g., regularization, early stopping, dropout, scheduling)	$\bullet$ Choice of the activation functions $\bullet$ Model input variables
CANN and GI-CANN	Various (usually gradient-based) $\bullet$ SGD $\bullet$ Adam $\bullet$ L-BFGS	$\bullet$ Learning rate $\eta$ $\bullet$ Number of epochs $\bullet$ Number of layers $\bullet$ Number of neurons per layer $\bullet$ Optional ML hyperparameters (e.g., regularization, early stopping, dropout, scheduling)	$\bullet$ Choice of the activation functions $\bullet$ Model input variables

Table 2: Hyperparameters of machine learning methods for constitutive modeling.

Figure 3: Method-specific core runtime of the six methods over 100 independent runs. Bars represent the arithmetic mean and error bars indicate one standard deviation. The core runtime comprises the Coordinate Descent solve for EUCLID, pattern recognition for Material Fingerprinting (MF), the adaptive loop for Adaptive Material Fingerprinting (AMF), and optimizer training after JAX warm-up for PANN, CANN, and GI-CANN. The runtime is reported in seconds on a logarithmic scale.

Figure 4: Mean squared error (MSE) of the six methods for uniaxial tension (UT), equibiaxial tension (ET), and pure shear (PS). Bars represent the arithmetic mean and error bars indicate one standard deviation. EUCLID and Material Fingerprinting (MF) each contribute one deterministic result; Adaptive Material Fingerprinting (AMF) comprises 20 combinations of the adaptive parameters

N_{a}

and

s

; and PANN, CANN, and GI-CANN each comprise 100 model initializations using different random seeds. Lower values indicate better predictive performance.

Figure 5: Coefficient of determination

R^{2}

of the six methods for uniaxial tension (UT), equibiaxial tension (ET), and pure shear (PS). Bars represent the arithmetic mean and error bars indicate one standard deviation. EUCLID and Material Fingerprinting (MF) each contribute one deterministic result; Adaptive Material Fingerprinting (AMF) comprises 20 combinations of the adaptive parameters

N_{a}

and

s

; and PANN, CANN, and GI-CANN each comprise 100 model initializations using different random seeds. Values closer to one indicate better predictive performance.

Figure 6: Predictive and training comparison of the six methods. Rows correspond to EUCLID, Material Fingerprinting (MF), Adaptive Material Fingerprinting (AMF), PANN, CANN, and GI-CANN. The first three columns show the first Piola–Kirchhoff stress component $P_{11}$ as a function of the stretch ratio $\lambda$ for uniaxial tension (UT), equibiaxial tension (ET), and pure shear (PS), respectively. Black markers denote the experimental reference data, solid lines denote the mean predictions, and shaded regions indicate one standard deviation. For AMF, the variation reflects the investigated 20 hyperparameter configurations, whereas for PANN, CANN, and GI-CANN it reflects 100 model initializations using different random seeds. The fourth column shows the individual training loss histories on a logarithmic scale. No iterative loss history is available for EUCLID, MF, or AMF.

Figure 7: Strain energy density

W(\bm{F})

predicted by the selected best-performing models: EUCLID, Material Fingerprinting (MF), Adaptive Material Fingerprinting (AMF), PANN, CANN, and GI-CANN. Each panel shows

W

as a function of the two independent principal stretches

\lambda_{1}

and

\lambda_{2}

, with the third stretch determined by the incompressibility constraint

\lambda_{3}=(\lambda_{1}\lambda_{2})^{-1}

. Identical contour levels and a common color scale are used in all six panels, enabling a direct comparison of the predicted energy landscapes. The marker at

(\lambda_{1},\lambda_{2})=(1,1)

denotes the undeformed reference configuration.

Figure 8: Runtime comparison for evaluating the strain energy density function

W

, its gradient

\bm{P}

, and its Hessian

\mathbb{H}

for the six methods. Bars show the mean runtime per deformation gradient and error bars indicate one standard deviation over 100 repetitions after two warm-up runs. All evaluations were performed in double precision on the CPU using batches of 100 deformation gradients. The vertical axis uses a logarithmic scale. Compilation and other initialization steps are excluded from the reported runtimes. We note that for EUCLID and the neural network-based methods, we measured the evaluation times for the full model and did not exclude features that were identified as inactive during the inverse problem. Excluding such features could further reduce evauation times.

Hyperelasticity

Hyperelasticity models large, reversible deformations in materials using nonlinear stress–strain laws, supporting simulation of soft solids and elastomers.

See all 18 papers

May 22, 2026

Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Data

The identification of constitutive neural network models from heterogeneous full-field deformation data provides a robust alternative to traditional calibration methods based on homogeneous stress-strain experiments, particularly given the high dimensionality of trainable parameters. This work presents ADE-FEMU (Automatic-Differentiation-Enabled Finite Element Model Updating), an end-to-end differentiable framework that integrates a constitutive neural network model within a JAX-FEM nonlinear solver and identifies its parameters through gradient-based minimization of a measurement-mismatch loss, allowing for training on structural data with limited kinetic information. Thus, it can be considered as an extension of finite element model updating (FEMU) for hyperelastic materials, implemented within a GPU-accelerated, fully automatically differentiable finite element framework Newton tangent stiffness and loss gradients are computed automatically using forward- and reverse-mode automatic differentiation throughout the entire pipeline, thereby removing the need for analytic adjoints or offline surrogate models. ADE-FEMU is demonstrated for two architectures: a grey-box Constitutive Artificial Neural Network (CANN), a polyconvex, fully connected model with high flexibility, and a white-box CANN, an expert-system network with phenomenologically interpretable strain-energy terms. Focusing on incompressible isotropic hyperelasticity, ADE-FEMU is evaluated on three open experimental datasets: (1) full digital image correlation (DIC) of a perforated tensile specimen, (2) a reduced-data scenario with a one-dimensional stretch profile and global force-displacement curve, and (3) a heterogeneous matrix-inclusion system in which both phases constitutive laws are identified and generalized to twenty-two previously unseen samples.

Matthias Knipper, Chenyi Ji, Malte Brand +1

Automatic Differentiation Hyperelasticity Materials Science Physics-Informed ML Finite Element Method

Ask PDF

Mar 27, 2026

Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networks

The discovery of constitutive laws from experimentally accessible measurements is a central problem in nonlinear computational mechanics. Many data-driven constitutive identification approaches rely either on paired strain-stress data or on full-field displacement measurements, both of which are difficult to obtain in realistic three-dimensional settings. We present a differentiable finite element framework for the discovery of hyperelastic material laws from partial observations, including boundary-only displacement measurements and global reaction forces. The method embeds the nonlinear finite element equilibrium problem directly into the learning loop, so that candidate strain-energy densities are assessed through the deformation fields and reactions they induce. This formulation enforces mechanical equilibrium as a constraint and allows the loss function to be evaluated only at observed locations. To ensure physical admissibility and promote numerical solvability throughout training, the constitutive response is represented by Hyperelastic Neural Networks, a structure-preserving neural class that enforces residual energy and stress-free conditions, frame indifference, isotropic material symmetry, polyconvexity, coercivity, and controlled volumetric growth by construction. The resulting PDE-constrained learning problem is solved using a quasi-Newton strategy combined with continuation and solver-aware backtracking. Numerical experiments in two- and three-dimensional finite elasticity demonstrate accurate recovery of hyperelastic isotropic responses from boundary-only data, robustness to measurement noise, and generalization across geometries, loading conditions, and boundary conditions.

Francesco Regazzoni

MOX, Department of Mathematics, Politecnico di Milano, Milan, Italy

Structure-Preserving Neural Networks Hyperelasticity Physics-Informed ML Differentiable Physics

Ask PDF

Feb 19, 2026

Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models

A key problem of solid mechanics is the identification of the constitutive law of a material, that is, the relation between strain history and stress. Machine learning has lead to considerable advances in this field lately. Here we introduce inelastic Constitutive Kolmogorov-Arnold Networks (iCKANs). This novel artificial neural network architecture can discover in an automated manner symbolic constitutive laws describing both the elastic and inelastic behavior of materials. That is, it can translate data from material testing into corresponding elastic and inelastic potential functions in closed mathematical form. We demonstrate the advantages of iCKANs using both synthetic data and experimental data of the viscoelastic polymer materials VHB 4910 and VHB 4905. The results demonstrate that iCKANs accurately capture complex viscoelastic behavior while preserving physical interpretability. It is a particular strength of iCKANs that they can process not only mechanical data but also arbitrary additional information available about a material (e.g., about temperature-dependent behavior). This makes iCKANs a powerful tool to discover in the future also how specific processing or service conditions affect the properties of materials.

Chenyi Ji, Kian P. Abdolazizi, Hagen Holthusen +2

Computational Mechanics in Medicine, Applied Medical Engineering, RWTH Aachen University Pauwelsstraße 20, 52074 Aachen, Germany · Institute for Continuum and Material Mechanics, Hamburg University of Technology Eißendorfer Straße 42, 21073 Hamburg, Germany · Institute of Applied Mechanics, University of Erlangen-Nuremberg Egerlandstrasse 5, 91058 Erlangen, Germany +1

Materials Science Neural Network Interpretability Kolmogorov-Arnold Networks Computational Materials Science Symbolic Regression

Ask PDF

Abstract

Hyperelasticity

Explore similar work

Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Data

Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networks

Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models