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

TL;DR

This paper introduces a novel Bayesian framework for uncertainty quantification in material model discovery, leveraging Gaussian processes and normalizing flows to handle noisy data and complex parameter distributions.

Contribution

It proposes a flexible, prior-free method that combines Gaussian processes and normalizing flows for uncertainty quantification and interpretable model discovery in material science.

Findings

Effective in isotropic and anisotropic data scenarios

Handles complex, non-linear parameter distributions

Provides sparse, interpretable constitutive models

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 linear coefficients of the model library or are limited in the flexibility of the inferred parameter probability distribution. We therefore propose a 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 constitutive models with inner-non-linear parameters: 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 modeling 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 experimental anisotropic data.