Heteroscedastic Gaussian Process Regression for material structure–property relationship modeling

Uncertainty quantification is a critical aspect of machine learning models for material property predictions. Gaussian Process Regression (GPR) is a popular technique for capturing uncertainties, but most existing models assume homoscedastic aleatoric uncertainty (noise), which may not adequately represent the heteroscedastic behavior observed in real-world datasets. Heteroscedasticity arises from various factors, such as measurement errors and inherent variability in material properties. Ignoring heteroscedasticity can lead to lower model performance, biased uncertainty estimates, and inaccurate predictions. Existing Heteroscedastic Gaussian Process Regression (HGPR) models often employ complicated structures to capture input-dependent noise but may lack interpretability. In this paper, we propose an HGPR approach that combines GPR with polynomial regression-based noise modeling to capture and quantify uncertainties in material property predictions while providing interpretable noise models. We demonstrate the effectiveness of our approach on both synthetic and physics-based simulation datasets, including mechanical properties (effective stress) of porous materials. We also introduce an approximated expected log predictive density method for model selection, which eliminates the need for retraining the model during leave-one-out cross-validation, allowing for efficient hyperparameter tuning and model evaluation. By capturing heteroscedastic behavior, enhancing interpretability, and improving model selection, our approach contributes to the development of more robust and reliable machine learning models for material property predictions, enabling informed decision-making in material design and optimization.