Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID

TL;DR

This work adapts the EUCLID framework to supervised discovery of interpretable hyperelastic constitutive laws for human brain tissue using labeled uniaxial and torsion data. By constructing a large, expressive model library that combines Mooney–Rivlin and Ogden-like features and applying sparse, nonnegative regression with Pareto analysis and clustering, the method identifies concise symbolic energy densities that fit experimental measurements with quantifiable accuracy. Key findings show that a one-term Ogden model suffices for the majority of specimens, with notable instances of multi-term Ogden or Mooney–Ogden combinations, and mean errors on the brain data are modest (average $\overline{\mathrm{MSE}} \approx 0.17$). This approach provides interpretable, data-driven constitutive laws suitable for brain-mechanics applications, while highlighting limitations due to data labeling, geometry nonidealities, and isotropy assumptions, and outlining future extensions to unlabeled data and more complex material behavior.

Abstract

We propose an automated computational algorithm for simultaneous model selection and parameter identification for the hyperelastic mechanical characterization of human brain tissue. Following the motive of the recently proposed computational framework EUCLID (Efficient Unsupervised Constitutive Law Identitication and Discovery) and in contrast to conventional parameter calibration methods, we construct an extensive set of candidate hyperelastic models, i.e., a model library including popular models known from the literature, and develop a computational strategy for automatically selecting a model from the library that conforms to the available experimental data while being represented as an interpretable symbolic mathematical expression. This computational strategy comprises sparse regression, i.e., a regression problem that is regularized by a sparsity promoting penalty term that filters out irrelevant models from the model library, and a clustering method for grouping together highly correlated and thus redundant features in the model library. The model selection procedure is driven by labelled data pairs stemming from mechanical tests under different deformation modes, i.e., uniaxial compression/tension and simple torsion, and can thus be interpreted as a supervised counterpart to the originally proposed EUCLID that is informed by full-field displacement data and global reaction forces. The proposed method is verified on synthetical data with artificial noise and validated on experimental data acquired through mechanical tests of human brain specimens, proving that the method is capable of discovering hyperelastic models that exhibit both high fitting accuracy to the data as well as concise and thus interpretable mathematical representations.

Figures (10)

• Figure 1: Qualitative illustration of the Pareto analysis for selecting an appropriate value of $\lambda_p$.
• Figure 2: Qualitative illustration of the feature clustering.
• Figure 3: Stretch versus stress response under uniaxial compression/tension and (normalized) twist versus torque response under simple torsion of the discovered material models in comparison with the synthetical data (noise level $\sigma = 10 \ \text{Pa}$).
• Figure 4: Illustration of the Pareto analysis for benchmark case MR2O1 (noise level $\sigma = 10 \ \text{Pa}$).
• Figure 5: Illustration of the feature clustering for benchmark case MR2O1 (noise level $\sigma = 10 \ \text{Pa}$).