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

2025-10-25

AI Summary

This study addresses the challenge of uncertainty quantification in constitutive model discovery—specifically, when no prior assumptions about model parameters are available. We propose a four-step semi-Bayesian framework: (1) Gaussian process regression to denoise and augment stress–strain data; (2) nonparametric approximation of the joint posterior parameter distribution via normalizing flows; (3) model structure identification and distillation through functional-space distribution matching; and (4) enhanced interpretability using Sobol’ sensitivity analysis. Crucially, the method requires no prespecified parameter priors, handles both linear and nonlinear model libraries uniformly, and automatically discovers interpretable constitutive forms. It significantly improves accuracy in uncertainty propagation. The framework is validated on both isotropic and anisotropic experimental datasets, demonstrating robust performance across diverse material behaviors.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Quantifying uncertainty in material model discovery from noisy mechanical test data
• Overcoming limitations in prior parameter selection and distribution flexibility
• Discovering interpretable nonlinear constitutive models using Bayesian framework

Innovation

• Uses Gaussian process to augment stress-deformation data
• Approximates parameter distribution via normalizing flow
• Distills interpretable models through sensitivity analysis