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.

中文翻译：

---

用于材料结构-性能关系建模的异方差高斯过程回归


不确定性量化是材料性能预测机器学习模型的一个关键方面。高斯过程回归 (GPR) 是一种捕获不确定性的流行技术，但大多数现有模型都假设同方差任意不确定性（噪声），这可能无法充分代表在现实世界数据集中观察到的异方差行为。异方差性由多种因素引起，例如测量误差和材料属性的固有变异性。忽略异方差性可能会导致模型性能降低、不确定性估计有偏差和预测不准确。现有的异方差高斯过程回归 (HGPR) 模型通常采用复杂的结构来捕获与输入相关的噪声，但可能缺乏可解释性。在本文中，我们提出了一种 HGPR 方法，它将 GPR 与基于多项式回归的噪声建模相结合，以捕获和量化材料特性预测中的不确定性，同时提供可解释的噪声模型。我们证明了我们的方法在合成和基于物理的模拟数据集上的有效性，包括多孔材料的机械性能（有效应力）。我们还引入了一种用于模型选择的近似预期对数预测密度方法，该方法消除了在留一交叉验证期间重新训练模型的需要，从而实现高效的超参数调整和模型评估。通过捕获异方差行为、增强可解释性和改进模型选择，我们的方法有助于开发更强大、更可靠的机器学习模型来预测材料性能，从而在材料设计和优化中做出明智的决策。