# Unsupervised discovery of interpretable hyperelastic constitutive laws

arXiv: https://arxiv.org/abs/2010.13496 · HTML: https://mlsanity.com/paper/2010.13496/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2020.

## Built on by (most cited first, 30 of 51)

- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics](https://mlsanity.com/paper/2310.03652/) (2023)
- [A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity](https://mlsanity.com/paper/2204.12088/) (2022)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries](https://mlsanity.com/paper/2110.01359/) (2021)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimes](https://mlsanity.com/paper/2305.03184/) (2023)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [G2Phinet: Relating Genotype and Biomechanical Phenotype of Tissues with Deep Learning](https://mlsanity.com/paper/2208.09889/) (2022)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling](https://mlsanity.com/paper/2410.13645/) (2024)
- [Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models](https://mlsanity.com/paper/2602.17750/) (2026)
- [Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanics](https://mlsanity.com/paper/2212.01386/) (2022)
- [DCEM: A deep complementary energy method for solid mechanics](https://mlsanity.com/paper/2302.01538/) (2023)
- [On the definition and importance of interpretability in scientific machine learning](https://mlsanity.com/paper/2505.13510/) (2025)
- [Neural Networks for Generating Better Local Optima in Topology Optimization](https://mlsanity.com/paper/2407.17957/) (2024)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials](https://mlsanity.com/paper/2508.12063/) (2025)
- [Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteria](https://mlsanity.com/paper/2509.16040/) (2025)
- [Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks](https://mlsanity.com/paper/2405.18311/) (2024)

## Abstract (arXiv)

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 l_p 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.