Adaptive Material Fingerprinting for the fast discovery of polyconvex feature combinations in isotropic and anisotropic hyperelasticity

Abstract: We recently proposed a method called Material Fingerprinting for the rapid discovery of mechanical material models that avoids solving continuous optimization problems. Material Fingerprinting assumes that each material exhibits a unique response when subjected to a standardized experimental setup, which is interpreted as the material's mechanical fingerprint. If a database of fingerprints is generated in an offline phase, a model for an unseen experimental measurement can be discovered in real time by comparing the experimentally measured fingerprint to the fingerprints in the database. In our original contributions, the database comprised a fixed number of material models, each with a fixed number of parameters. To increase the fitting flexibility of Material Fingerprinting, we propose an adaptive model database coupled with an iterative pattern recognition algorithm that refines the material model in each step. This strategy enables Material Fingerprinting to discover arbitrary linear combinations of material models from the database, rather than being restricted to selecting a single model from a predefined set. In comparison to previous works on Material Fingerprinting, this enables the discovery of more complex models, such as multi-term Ogden models or the anisotropic Holzapfel-Gasser-Ogden model. To design the adaptive database, we leverage sums of strain energy density feature functions that depend on isotropic and anisotropic invariants. All modeling features satisfy fundamental physical constraints, and polyconvexity can be optionally enforced via a simple user-controlled switch. We test the method on experimental data stemming from mechanical tests of isotropic rubber materials and anisotropic animal skin tissue.

• The paper introduces Adaptive Material Fingerprinting (AMF) for fast, interpretable discovery of polyconvex constitutive models in hyperelastic solids.
• It employs a combinatorial database of parametrized, physically-constrained feature functions and an iterative pattern recognition algorithm to build sparse models.
• Numerical experiments on isotropic rubber and anisotropic skin demonstrate high accuracy (R² up to 0.99) while preserving essential physical properties.

Adaptive Material Fingerprinting for Polyconvex Feature Discovery in Hyperelasticity

Overview

The paper "Adaptive Material Fingerprinting for the fast discovery of polyconvex feature combinations in isotropic and anisotropic hyperelasticity" (2604.05698) introduces an adaptive, combinatorial approach to material model discovery for hyperelastic solids. The method, termed Adaptive Material Fingerprinting (AMF), is an extension of previous lookup-table fingerprinting methodologies, designed to enable efficient and interpretable identification of complex, physically-admissible constitutive models without continuous optimization. Major advances include a feature-combinatorial search algorithm, a database of parametric physically-constrained features (admitting polyconvexity), and an iterative pattern recognition framework for model construction.

Figure 1: Workflow of adaptive Material Fingerprinting: an offline phase computes feature model responses and stores them, while an online phase iteratively constructs a sparse model to fit experimental fingerprints.

Motivation and Context

Traditional parameter identification in nonlinear elasticity relies on the selection of a constitutive model and a subsequent (potentially nonconvex and high-dimensional) optimization to fit experimental data. Recent advances in machine learning for constitutive modeling—such as neural network surrogates and symbolic regression—alleviate the need for a priori selection of model form, offering flexibility but often at the cost of interpretability, physical admissibility, and computational efficiency. These ML models typically require stress-strain data and expensive training processes.

The original Material Fingerprinting approach circumvented continuous optimization by working with a precomputed database of material response “fingerprints,” but was limited to fixed model sets. Adaptive Material Fingerprinting lifts this restriction by allowing linear combinations of database entries, thus enabling discovery of multi-feature, physically constrained models.

Methodology

Database Construction

Feature functions—forming the basis of candidate strain energies—are constructed using invariants of deformation, including principal stretches/areas for isotropy and fiber-directional quantities for anisotropy. The precomputed database stores the stress response ("fingerprint") of each feature function evaluated across a range of deformation protocols (uniaxial, shear, equibiaxial, etc.), with relevant parameter sweeps.

Figure 2: Standardized deformation protocols used for fingerprinting, parameterized by principal stretches under incompressibility constraints.

Specific forms for the isotropic and anisotropic features use well-known classes of functions (e.g., powers, exponential, softplus-type) that subsume classical models such as the Ogden and Holzapfel-Gasser-Ogden formulations, with parameter ranges chosen to ensure a flexible and rich basis. For polyconvexity, parameter constraints are enforced such that feature convexity in singular values is guaranteed.

Pattern Recognition and Model Building

Material discovery is treated as an adaptive, greedy feature selection problem. After experimental stress data is interpolated and normalized to match the database deformation grid, AMF iteratively constructs a model:

1. Initialization: The strain energy density starts as zero.
2. Residual Computation: At each step, the residual between the experimental response and the current model is computed.
3. Feature Selection: The database entry whose fingerprint best matches the normalized residual is identified (maximal inner product).
4. Model Update: The selected feature is added to the strain energy, weighted by a determined scaling parameter.
5. Iteration: Steps 2–4 repeat for a user-prescribed number of features ($n_a$), controlling model sparsity.

A key hyperparameter $s$ modulates the update strength at each step. The final model is a sum of up to $n_a$ feature terms, yielding a sparse, interpretable constitutive law.

Numerical and Experimental Results

Isotropic Rubber

Experiments on Treloar rubber data at two temperatures showcase the progression of model fit as $n_a$ increases:

• $n_a=1$:
  • Single-feature (exponential stretch) model captures qualitative trends with $R^2=0.93$.
• $n_a=3$ and above:
  • Multi-term models accurately describe all deformation modes, approaching $R^2 > 0.99$.
  • Discovered terms align with Ogden-type and area-based features.
  • 

  • Figure 3: Stress-stretch data and 3-term discovered model for rubber at $20^{\circ}\mathrm{C}$. Successive feature addition improves the fit for all test protocols.

Anisotropic Skin

Application to porcine skin data demonstrates necessity of anisotropic features:

• $n_a=1$:
  • Isotropic-only model yields poor fit ($s$0).
• $s$1:
  • The addition of a single anisotropic term (fiber stretch-based feature) improves $s$2.
• $s$3 or more:
  • Further terms enhance fit ($s$4), with diminishing returns and model sparsity easily controllable by $s$5.
  • 

  • Figure 4: Stress-stretch data and discovered model for anisotropic skin, highlighting the contribution of anisotropic features in accurately capturing $s$6 under fiber direction loading.

Hyperparameter Sensitivity and Polyconvexity

Systematic scanning reveals that moderate $s$7 and $s$8–$s$9 balances accuracy and sparsity. Enforcing polyconvexity yields no loss in fit for rubber data; for skin, a small decrease in $n_a$0 is observed when polyconvex features are enforced, supporting the practical utility of the approach for robust model discovery.

Figure 5: Hyperparameter heatmap for $n_a$1 and $n_a$2 showing their effect on goodness of fit for skin tissue data.

Theoretical and Practical Implications

Theoretical Advantages

The AMF framework retains physical admissibility—objectivity, stress-free reference, symmetry, and polyconvexity—by feature construction and database constraint. The pattern recognition-based selection obviates the need for nonlinear, potentially ill-posed optimization, and always yields a valid model from the candidate set.

By mimicking forward stepwise regression but leveraging a database of physically-constrained, parametrized features, AMF avoids overfitting and provides transparent trade-offs between sparsity and fit. The explicit, interpretable functional forms discovered are distinct from typical “black-box” ML surrogates and aid mechanism identification.

Comparison to Machine Learning Methods

Direct comparison to ANN-based constitutive modeling [martonova_generalized_2025], [linka_automated_2023-1] shows that AMF achieves similar $n_a$3 values without the overhead of training or the interpretability problems of neural networks. Unlike CANNs or other surrogate models, AMF does not require stress labels and is compatible with “physics-only” fingerprints.

Practical Utility

AMF is suitable for rapid experimental model identification, especially in contexts where interpretability, thermodynamic consistency, or computational efficiency are required (e.g., automated characterization in biomedical or soft robotics, digital twin applications). The separation into offline/online stages makes the approach scalable and reusable.

Future Outlook

Further extensions include enlarging the fingerprint database to cover a richer set of features (e.g., incorporating strain-gradient or viscoelastic terms), automated hyperparameter tuning for unsupervised model complexity selection, and end-to-end integration with digital image correlation or full-field experimental data (potentially extending beyond homogeneous deformations).

Integration with symbolic regression and Bayesian uncertainty estimation for model selection may enable further interpretability and robustness advances, enhancing predictive capabilities in material informatics workflows.

Conclusion

Adaptive Material Fingerprinting (2604.05698) provides a robust, interpretable, and physically-constrained framework for data-driven constitutive model discovery. Its iterative, combinatorial, and database-driven architecture enables discovery of sparse, accurate models efficiently, sidestepping the nonconvex optimization bottleneck of traditional and machine learning approaches. Applications to complex, anisotropic, and polyconvex materials underline its versatility and potential as a reliable tool in automated material characterization and computational mechanistic modeling.