Unsupervised discovery of interpretable hyperelastic constitutive laws

Moritz Flaschel, Siddhant Kumar, Laura De Lorenzis*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

65 Citations (Scopus)
363 Downloads (Pure)


We propose a new approach for data-driven automated discovery of isotropic hyperelastic constitutive laws. The approach is unsupervised, i.e., it requires no stress data but only displacement and global force data, which are realistically available through mechanical testing and digital image correlation techniques; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a large catalogue of candidate functions; it is one-shot, i.e., discovery only needs one experiment — but can use more if available. The problem of unsupervised discovery is solved by enforcing equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity of the solution is achieved by ℓp regularization combined with thresholding, which calls for a non-linear optimization scheme. The ensuing fully automated algorithm leverages physics-based constraints for the automatic determination of the penalty parameter in the regularization term. Using numerically generated data including artificial noise, we demonstrate the ability of the approach to accurately discover five hyperelastic models of different complexity. We also show that, if a “true” feature is missing in the function library, the proposed approach is able to surrogate it in such a way that the actual response is still accurately predicted.

Original languageEnglish
Article number113852
Number of pages19
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 2021


  • Constitutive models
  • Hyperelasticity
  • Interpretable models
  • Inverse problems
  • Sparse regression
  • Unsupervised learning

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