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Benchmarking data-driven material models on the classic Treloar dataset

Hagen Holthusen, Moritz Flaschel, Denisa Martonová, Ellen Kuhl

arXiv:2608.14063v1cs.AIcs.CE

TL;DR

Machine-learning constitutive models are increasingly diverse, but their practical performance and complexity are difficult to compare. This benchmark evaluates six approaches on Treloar’s rubber data and finds complementary strengths rather than a single universally optimal method.

  • Problem

    The study addresses limited systematic evidence comparing data-driven hyperelasticity methods across predictive accuracy, complexity, computational cost, and practical selection criteria.

  • Method

    The authors benchmark six data-driven constitutive modeling approaches on Treloar’s rubber dataset using a common training and validation setup.

  • Results

    All six methods reproduce the experimental responses accurately, while AMF and GI-CANN lead selected-model accuracy and EUCLID and MF reduce identification cost and representation sparsity.

  • Takeaways & Limitations

    Method choice should reflect whether predictive accuracy, sparsity, robustness, or computational efficiency is the application’s highest priority.

  • Takeaways & Limitations

    The benchmark reports MF’s supervised online cost but excludes its one-time database-generation effort.

Abstract

from arXiv · show

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.

1. Introduction

The introduction positions Treloar’s natural-rubber experiments as a central benchmark for constitutive-model development and motivates a systematic comparison of representative data-driven methods. The study evaluates fitting accuracy alongside model complexity and computational cost, while explicitly limiting its scope to selected widely used approaches rather than claiming a universal winner.

  • Motivation: Treloar’s natural-rubber experiments are among the most widely used benchmarks for developing and evaluating constitutive models.Earlier work compared fourteen phenomenological and micromechanically motivated hyperelastic models and tested cross-mode predictive capability.
  • Evaluation criteria: The comparison evaluates data-driven methods using complementary metrics for fitting accuracy, model complexity, and inverse-problem computational cost.The study introduces several fitting metrics and investigates the computational cost of the considered methods.
  • Evaluation criteria: Constitutive-model complexity matters because simulations repeatedly evaluate models and their derivatives at every integration point.This repeated evaluation can substantially affect overall computational cost.
  • Scope and limitations: The study compares a representative set of widely used approaches and does not claim to cover all state-of-the-art methods or identify a universally best-performing technique.The introduction specifically includes Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) and Material Fingerprinting (MF) among the considered approaches.

2. Benchmark problem

The benchmark uses Treloar’s rubber data across three deformation modes, with uniaxial and equibiaxial tension for calibration and pure shear held out for validation. Models are trained on normalized nominal stresses and assessed by fit, computational cost, and resulting model complexity.

  • Benchmark dataset: Treloar’s 20°C rubber dataset contains stretch–nominal stress measurements for uniaxial tension, equibiaxial tension, and pure shear.The measured stretches and P11 stresses are converted into three-dimensional deformation gradients under incompressibility.
  • Data split: Uniaxial and equibiaxial data form the calibration set, while pure shear is excluded from training as an independent validation case.This split tests generalization to a deformation mode absent from calibration.
  • Preprocessing: Nominal stress targets are normalized by the training-set maximum absolute stress, then predictions are returned to physical units without smoothing, filtering, or augmentation.Normalization improves numerical stability while preserving direct comparison with the original measurements.
  • Training objective: The supervised objective minimizes mean squared error between normalized experimental and predicted P11 stresses after incompressible pressure correction enforcing P33 = 0.The correction removes the hydrostatic contribution and evaluates the physically admissible stress response.
  • Evaluation criteria: Methods are compared by identification time, goodness of fit, interpretability, and simplicity, including fit metrics and effective nonzero parameter count ∥θ∥0.Evaluation also considers constitutive evaluation cost because equal parameter counts can entail different computational workloads.

3. Methods

The methods construct representations of the hyperelastic strain-energy density for isotropic incompressible materials, using invariant-based formulations and approaches ranging from sparse discovery to neural networks. They differ in how they select constitutive representations, optimize or search model spaces, and expose modeling choices and hyperparameters.

  • Common formulation: All methods target a mathematical representation of the strain-energy density for isotropic incompressible hyperelasticity, commonly expressed using invariants of C = F^T F.The incompressibility constraint is J = 1, enforced through a scalar Lagrange multiplier.
  • EUCLID: EUCLID uses sparse regression over a prescribed library of modeling terms to discover interpretable material models with few terms from data.Its performance depends on the library size, regularization α, threshold θ→0, feature variables, and included functional forms.
  • Material Fingerprinting: Material Fingerprinting replaces continuous optimization with database pattern recognition, using matrix–vector multiplication and a maximum search to identify the discrete global optimum in the searchable model space.Its database requires a priori choices of variables, admissible functional forms, and parameter-sampling density.
  • Adaptive Material Fingerprinting: Adaptive Material Fingerprinting combines multiple features rather than selecting one predefined model, retaining database-search efficiency while producing sparse, physically interpretable models without continuous optimization.The number of included terms Na and step size s are specified a priori and can be tuned by grid search.
  • Neural-network methods: PANNs use trainable neural-network parameters to model the strain energy, while GI-CANNs additionally train the invariant representation through continuous generalized-invariant exponents.PANN design and training require choices such as architecture, inputs, learning rate, and epochs; GI-CANNs recover standard- and principal-stretch-based CANN representations as special cases.

4. Results

All six methods reproduce Treloar’s stress–stretch responses accurately, while differing in deformation-mode performance, robustness, complexity, computational cost, and extrapolated energy landscapes. EUCLID and GI-CANN offer strong accuracy with sparse representations, whereas neural-network methods generally require substantially higher identification costs.

  • Accuracy: All six methods closely reproduce experimental responses across uniaxial tension, equibiaxial tension, and pure shear, with differences emerging mainly between deformation modes.EUCLID achieves R2 = 0.9985 in uniaxial tension and 0.9978 in pure shear, while GI-CANN reaches R2 = 0.9978 in equibiaxial tension.
  • Accuracy: AMF reaches the highest selected-model validation accuracy at R2 = 0.9996, while GI-CANN attains training R2 = 0.9996 and validation R2 = 0.9993.The representative models were selected using minimum validation MSE in the testing pure shear mode.
  • Accuracy–complexity trade-off: MF uses two nonzero parameters, EUCLID four, GI-CANN eight, CANN 12, AMF 49, and PANN 371, with PANN lacking a corresponding validation-accuracy increase.For Na = 20, AMF has the highest selected-model validation accuracy but requires a substantially larger representation than EUCLID or GI-CANN.
  • Robustness: PANN shows the greatest initialization sensitivity, with mean R2 = 0.9039 and standard deviation 0.2450 in equibiaxial tension, versus GI-CANN’s R2 = 0.9978 ± 0.0035.EUCLID and MF are deterministic for the selected settings, whereas AMF varies across Na and s combinations and PANN, CANN, and GI-CANN vary across random initializations.
  • Computational cost: EUCLID and MF require mean core times of 0.337 ms and 0.305 ms, respectively, while CANN, PANN, and GI-CANN require 324.539 ms, 588.495 ms, and 681.277 ms.MF’s reported core time excludes offline database generation; AMF adds adaptive feature search but remains faster than neural-network methods.
  • Evaluation and extrapolation: MF has the lowest evaluation times, requiring 0.151 µs for W, 0.473 µs for P, and 3.263 µs for H per deformation gradient.The six models produce smooth energy landscapes with minima at (λ1, λ2) = (1, 1); MF’s shifted contours correspond to larger equibiaxial deviations.

5. Conclusion

Benchmarking six data-driven constitutive modeling approaches on the Treloar rubber dataset reveals complementary trade-offs between predictive accuracy, identification cost, model sparsity, and evaluation cost. The results provide a quantitative basis for selecting methods according to application requirements.

  • Performance trade-offs: AMF and GI-CANN provide the highest predictive accuracy among the selected models, while EUCLID and MF offer substantially lower identification costs and sparser representations.GI-CANN combines high accuracy with a compact eight-parameter representation, whereas MF provides the lowest computational cost and sparsest model.
  • Performance trade-offs: GI-CANN combines high accuracy with a compact eight-parameter representation, while MF provides the lowest computational cost and sparsest model.These results illustrate complementary strengths among the benchmarked approaches.
  • Deployment considerations: Constitutive model evaluation cost can differ substantially from model identification cost and is therefore important for computational deployment.The benchmark distinguishes the cost of discovering or fitting a model from the cost of evaluating it in applications.
  • Deployment considerations: The benchmark provides a quantitative basis for selecting data-driven constitutive models according to the requirements of a specific application.Selection should account for the complementary strengths identified across the considered approaches.

Data and code availability

All datasets, numerical results, implementation files, and scripts required to reproduce the study are publicly available in a Zenodo repository. The archive includes implementations of all six investigated approaches and their training and evaluation scripts.

  • Data and code availability: The public archive contains all datasets, numerical results, and implementation files required to reproduce the analyses.It is hosted in a Zenodo repository.
  • Data and code availability: Implementations of the six investigated approaches are included in the repository.The archive provides the method implementations used in the study.
  • Data and code availability: The corresponding training and evaluation scripts are publicly deposited at https://doi.org/10.5281/zenodo.21915635.The repository contains the scripts needed to support reproducibility.

Appendix A. Technical details

Appendix A specifies the reproducible command-line settings, repeated evaluations, runtime-statistics procedure, computational hardware, and analytical principal-stretch evaluation used in the study.

  • Reproducibility: The comparison command uses 100 timing measurements, 100 neural-network seeds, EUCLID regularization α = 10^-5, and 100-sample, 100-repetition computational-cost studies.These settings correspond to the options --n-timing 100, --n-seeds 100, --euclid-lambdas 1e-5, --study4-num-samples 100, and --study4-repetitions 100.
  • Computational-cost evaluation: Each computational-cost study evaluates W, P = ∂W/∂F, and H = ∂2W/∂F∂F on a common set of 100 admissible incompressible deformation gradients.The complete evaluation set is repeated 100 times for every model–quantity combination.
  • Runtime statistics: Runtime per deformation gradient is obtained by dividing each repetition’s total runtime by 100, then computing the mean, median, and sample standard deviation across repetitions.Compilation and warm-up executions are excluded from these statistics, while cold-start time is recorded separately.
  • Computational environment: All reported runtime measurements use the CPU backend on an Apple M4 MacBook Pro with a 10-core CPU and 24 GB of unified memory running macOS 26.6.The processor has four performance cores and six efficiency cores.
  • Principal-stretch evaluation: For computational-cost evaluation, principal stretches are computed from the eigenvalues of the right Cauchy–Green tensor rather than by explicitly performing a singular-value decomposition.The symmetric 3 × 3 tensor’s eigenvalues are evaluated analytically using a closed-form trigonometric solution, with a separate near-isotropic treatment avoiding numerical degeneracy.
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