# Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties

arXiv: https://arxiv.org/abs/2203.07422 · HTML: https://mlsanity.com/paper/2203.07422/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2022.

## Built on by (most cited first, 23 of 23)

- [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)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (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)
- [Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen](https://mlsanity.com/paper/2308.11080/) (2023)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Discovering stochastic partial differential equations from limited data using variational Bayes inference](https://mlsanity.com/paper/2306.15873/) (2023)
- [Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteria](https://mlsanity.com/paper/2509.16040/) (2025)
- [Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)](https://mlsanity.com/paper/2604.07746/) (2026)
- [Investigating Deep Learning Model Calibration for Classification Problems in Mechanics](https://mlsanity.com/paper/2212.00881/) (2022)
- [Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study](https://mlsanity.com/paper/2510.09498/) (2025)
- [Automating modeling in mechanics: LLMs as designers of physics-constrained neural networks for constitutive modeling of materials](https://mlsanity.com/paper/2512.01735/) (2025)
- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)
- [Discovering uncertainty: Gaussian constitutive neural networks with correlated weights](https://mlsanity.com/paper/2503.12679/) (2025)
- [Learning Physics-Consistent Material Behavior from Dynamic Displacements](https://mlsanity.com/paper/2407.20273/) (2024)
- [Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics](https://mlsanity.com/paper/2501.00556/) (2024)
- [The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems](https://mlsanity.com/paper/2502.07415/) (2025)
- [LLM-driven design of physics-constrained constitutive models: two agents are better than one](https://mlsanity.com/paper/2605.23754/) (2026)
- [Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Data](https://mlsanity.com/paper/2606.05199/) (2026)
- [CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field data](https://mlsanity.com/paper/2606.14565/) (2026)
- [Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations](https://mlsanity.com/paper/2606.19375/) (2026)
- [Structure-preserving uncertainty quantification for GENERIC dynamics](https://mlsanity.com/paper/2608.12624/) (2026)

## Abstract (arXiv)

Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsity-promoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent-Thomas, Arruda-Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties - i.e. uncertainties on the true features of the constitutive catalog - and aleatoric uncertainties - which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.