Discovering uncertainty: Gaussian constitutive neural networks with correlated weights

TL;DR

This work introduces Gaussian constitutive neural networks to quantify material stress uncertainty while honoring physical constraints such as polyconvexity. By modeling external weights as correlated Gaussian variables and predicting both mean and variance of stresses, the approach eliminates the need for priors on weights and enables direct probabilistic predictions. Across three model classes (unregularized independent, regularized independent, and regularized correlated), the correlated variant delivers better stress-variance alignment with biaxial data and yields a compact, interpretable four-term energy model. The framework lays the groundwork for generative constitutive models and priors for new samples with limited data, offering a principled path toward uncertainty-aware, physics-consistent material modeling.

Abstract

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.

Figures (5)

• Figure 1: Gaussian constitutive neural network architecture. The first several layers of the network are identical to an ordinary constitutive neural network for anisotropic materials. The final layer has two sets of parameters, one which corresponds to the mean and one which corresponds to the variance. These terms are differentiated and summed together as summarized in Section \ref{['sec:arch']}.
• Figure 2: Experimental data from biaxial testing. The top two rows show experiments where the main or warp fiber direction is aligned with the loading direction, and the bottom row shows experiments where the main fiber direction is offset from the loading directions by 45 degrees. Each box represents a different experimental setup and shows data from all five samples.
• Figure 3: Experimental data and model prediction for unregularized, independent Gaussian constitutive neural network. The red line and shaded area show the mean and variance of the experimental data. The blue line and shaded area show the model predicted mean and variance. Within each plot, we report the extra NLL as the difference between the NLL and the minimum theoretical NLL given the data distribution, i.e., the NLL if the data and model distributions were identical.
• Figure 4: Experimental data and model prediction for regularized, independent Gaussian constitutive neural network with $\alpha = 0.10$. The red line and shaded area show the mean and variance of the experimental data. The blue line and shaded area show the model predicted mean and variance. Within each plot, we report the extra NLL as the difference between the NLL and the minimum theoretical NLL given the data distribution, i.e., the NLL if the data and model distributions were identical.
• Figure 5: Experimental data and model prediction for the regularized, correlated Gaussian constitutive neural network with $\alpha = 0.10$. The predicted s-stress in the strip-s and off-s experiments has a smaller variance, and thus matches the data distribution more closely than the previous models. The red line and shaded area show the mean and variance of the experimental data. The blue line and shaded area show the model predicted mean and variance. Within each plot, we report the extra NLL as the difference between the NLL and the minimum theoretical NLL given the data distribution, i.e., the NLL if the data and model distributions were identical.