Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteria

TL;DR

This work tackles automated discovery of constitutive laws for hyperelastic materials by pairing three sparse-regression methods (LASSO,LARS,OMP) with three model-selection criteria (AIC,BIC,CV), producing nine algorithm–criterion pipelines. The authors implement a library-based approach for both isotropic (Mooney–Rivlin and Ogden terms) and anisotropic (orthotropic with structural tensors) models, linearizing the non-linear parameters to form a regression problem and refining the selected terms via ridge or nonlinear least squares followed by hard-thresholding. Across synthetic benchmarks, Treloar data, and human cardiac tissue, all pipelines identify compact, physically consistent models with high predictive accuracy, while dramatically reducing computation time compared with prior nonlinear optimization approaches. The framework thus broadens the toolkit for data-driven constitutive-model discovery, enabling robust, automated identification of both isotropic and anisotropic hyperelastic materials suitable for engineering and biomechanics applications.

Abstract

The automated discovery of constitutive models from data has recently emerged as a promising alternative to the traditional model calibration paradigm. In this work, we present a fully automated framework for constitutive model discovery that systematically pairs three sparse regression algorithms Least Absolute Shrinkage and Selection Operator (LASSO), Least Angle Regression (LARS), and Orthogonal Matching Pursuit (OMP)) with three model selection criteria: $K$-fold cross-validation (CV), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). This pairing yields nine distinct algorithms for model discovery and enables a systematic exploration of the trade-off between sparsity, predictive performance, and computational cost. While LARS serves as an efficient path-based solver for the $\ell_1$-constrained problem, OMP is introduced as a tractable heuristic for $\ell_0$-regularized selection. The framework is applied to both isotropic and anisotropic hyperelasticity, utilizing both synthetic and experimental datasets. Results reveal that all nine algorithm-criterion combinations perform consistently well in discovering isotropic and anisotropic materials, yielding highly accurate constitutive models. These findings broaden the range of viable discovery algorithms beyond $\ell_1$-based approaches such as LASSO.

Figures (12)

• Figure 1: Isotropic Synthetic Data: Predicted stress-stretch responses for the best discovered models under 10% relative noise, corresponding to the data in Table \ref{['tab:synth_best_model_forms_kpa_nrmse_rel']}: (a) O2 ground truth, discovered by LASSO-AIC (2 terms). (b) MR2 ground truth, discovered by LASSO-CV (2 terms). (c) MR1O1 ground truth, discovered by LASSO-BIC (2 terms). (d) MR2O2 ground truth, discovered by LASSO-AIC (4 terms). Ground truth responses are shown for comparison.
• Figure 2: Isotropic Synthetic Data:. Model selection for sparse regression for MR2O2 benchmark at 10% relative noise. a. LASSO, b. LARS, and c. OMP. Each row displays a different selection criterion metric against either the regularization penalty term (for LASSO) or the number of steps (for LARS and OMP). Vertical dashed lines indicate the optimal complexity chosen by each criterion. The shaded area denotes the one-standard-error band for the CV curve calculated for the different CV-folds. NMSE stands for the normalized mean squared error.
• Figure 3: Isotropic Synthetic Data: LARS activation path for MR2 benchmark at 10% relative noise. The heatmap shows which model terms (y-axis) are active (blue) at each step of the algorithm (x-axis). The path illustrates the order in which terms are added to or removed from the model. Vertical lines mark the models selected by CV, AIC, and BIC.
• Figure 4: Isotropic Synthetic Data: OMP activation path for MR2O2 benchmark at 10% relative noise. The heatmap shows which model terms (y-axis) are active (blue) at each step of the algorithm (x-axis). The path illustrates the order in which terms are added to or removed from the model. Vertical lines mark the models selected by CV, AIC, and BIC.
• Figure 5: Isotropic Experimental Data - LASSO/OMP Model: Stress-stretch response of the four-term model discovered by LASSO and OMP methods compared against the Treloar experimental data. The model demonstrates excellent fidelity across the UT, PS, and EBT deformation modes, confirming its high predictive accuracy. The reported $R^2$ value in this figure corresponds to the overall value for the three deformation modes.