# Discovering uncertainty: Gaussian constitutive neural networks with correlated weights

arXiv: https://arxiv.org/abs/2503.12679 · HTML: https://mlsanity.com/paper/2503.12679/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 16 Mar 2025.

## Builds on (most cited first, 6 of 6)

- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties](https://mlsanity.com/paper/2203.07422/) (2022)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimes](https://mlsanity.com/paper/2305.03184/) (2023)
- [Exploring hyperelastic material model discovery for human brain cortex: multivariate analysis vs. artificial neural network approaches](https://mlsanity.com/paper/2310.10762/) (2023)

## Built on by (most cited first, 1 of 1)

- [Atrial constitutive neural networks](https://mlsanity.com/paper/2504.02748/) (2025)

## Abstract (arXiv)

When characterizing materials, it can be important to not only predict their mechanical properties, but also to estimate the probability distribution of these properties across a set of samples. Constitutive neural networks allow for the automated discovery of constitutive models that exactly satisfy physical laws given experimental testing data, but are only capable of predicting the mean stress response. Stochastic methods treat each weight as a random variable and are capable of learning their probability distributions. Bayesian constitutive neural networks combine both methods, but their weights lack physical interpretability and we must sample each weight from a probability distribution to train or evaluate the model. Here we introduce a more interpretable network with fewer parameters, simpler training, and the potential to discover correlated weights: Gaussian constitutive neural networks. We demonstrate the performance of our new Gaussian network on biaxial testing data, and discover a sparse and interpretable four-term model with correlated weights. Importantly, the discovered distributions of material parameters across a set of samples can serve as priors to discover better constitutive models for new samples with limited data. We anticipate that Gaussian constitutive neural networks are a natural first step towards generative constitutive models informed by physical laws and parameter uncertainty.