Uncertainty quantification in model discovery by distilling interpretable material constitutive models from Gaussian process posteriors

TL;DR

The paper addresses uncertainty quantification in constitutive model discovery from noisy stress–deformation data by proposing a four-step framework that augments data with a Gaussian process posterior, distills a flexible parameter distribution with a normalizing flow, and matches distributions of stress–deformation functions via Wasserstein-1 distance, followed by Sobol' sensitivity-based sparsification. This partially Bayesian approach avoids explicit priors over material parameters, supports linear and nonlinear model libraries, and yields a joint parameter distribution that preserves uncertainty while enforcing sparsity. Demonstrations on isotropic Treloar data and anisotropic cardiac tissue (including synthetic and experimental cases) show good mean fits and reasonable uncertainty estimates, with notes on limitations in anisotropic data due to library flexibility or fiber orientation. The work offers a practical path to interpretable, uncertainty-aware constitutive models and points to future enhancements such as monotone GP flows, non-stationary modeling, and extension to inelastic or unsupervised settings.

Abstract

Constitutive model discovery refers to the task of identifying an appropriate model structure, usually from a predefined model library, while simultaneously inferring its material parameters. The data used for model discovery are measured in mechanical tests and are thus inevitably affected by noise which, in turn, induces uncertainties. Previously proposed methods for uncertainty quantification in model discovery either require the selection of a prior for the material parameters, are restricted to the linear coefficients of the model library or are limited in the flexibility of the inferred parameter probability distribution. We therefore propose a four-step partially Bayesian framework for uncertainty quantification in model discovery that does not require prior selection for the material parameters and also allows for the discovery of non-linear constitutive models: First, we augment the available stress-deformation data with a Gaussian process. Second, we approximate the parameter distribution by a normalizing flow, which allows for capturing complex joint distributions. Third, we distill the parameter distribution by matching the distribution of stress-deformation functions induced by the parameters with the Gaussian process posterior. Fourth, we perform a Sobol' sensitivity analysis to obtain a sparse and interpretable model. We demonstrate the capability of our framework for both isotropic and anisotropic experimental data as well as linear and non-linear model libraries.

Figures (13)

• Figure 1: Workflow for the quantification of uncertainty in model discovery by distilling interpretable material constitutive models from GP posteriors.
• Figure 2: GP posterior for the Treloar dataset. The illustrations show the GP posterior mean, the centered 95%-intervals, some random stress-deformation function samples and the estimated coverages for the UT, EBT and PS test as well as the total estimated coverage. The GP posterior is used for data augmentation in the subsequent steps of the proposed framework.
• Figure 3: Distilled distribution over the material parameters after the sensitivity analysis and model refinement for the Treloar dataset. The unit of the linear parameters $c$ is $\unit{\kilo\pascal}$. The distribution over stress-deformation functions induced by this parameter distribution is shown in \ref{['fig:model_isotropic']}.
• Figure 4: Distilled interpretable statistical model for the Treloar dataset. The illustrations show the mean stress-deformation functions, the centered 95%-intervals, some random stress-deformation function samples as well as the individual and total validation metrics. The RMSE and the R2 refer to the mean stress-deformation functions, respectively, and show a good fit. Additionally, the results of the coverage estimation prove that the uncertainty is well estimated.
• Figure 5: Development of total Sobol' indices for the Treloar dataset over the course of the mechanical tests. The results show that the stress component $P _{11}$ predicted by the statistical model is most sensitive to the terms linearly parameterized in $c ^{(1,0)}$ (Neo-Hookean term), $c ^{(3,0)}$ and $c ^{(-1)}$. However, the effect of the terms differs for the different deformation modes and usually changes with increasing deformation.