标题： 基于机器学习预测的不确定性量化：用于联合模型和输入不确定性传播的多项式混沌展开方法 显示英文标题

标题： Uncertainty Quantification for Machine Learning-Based Prediction: A Polynomial Chaos Expansion Approach for Joint Model and Input Uncertainty Propagation

摘要： 机器学习（ML）代理模型在工程分析和设计中被越来越多地使用，以替代计算成本高昂的仿真模型，显著降低计算成本并加速决策过程。 然而，ML预测包含固有的误差，通常被估计为模型不确定性，这与模型输入的变异性相关联。 准确量化和传播这些综合不确定性对于生成可靠的工程预测至关重要。 本文提出了一种基于多项式混沌展开（PCE）的稳健框架，用于处理联合输入和模型不确定性传播。 虽然该方法广泛适用于一般的ML代理模型，但我们专注于高斯过程回归模型，这些模型为模型不确定性提供了明确的预测分布。 通过将所有随机输入转换为统一的标准空间，构建了一个PCE代理模型，使得输出的均值和标准差能够高效且准确地计算。 所提出的方法还提供了一种全局敏感性分析机制，能够准确量化输入变量和ML模型不确定性对整体输出变异性的个体贡献。 该方法为全面的不确定性量化提供了一个计算效率高且易于解释的框架，在后续的工程应用中支持可信赖的ML预测。 显示英文摘要

摘要： Machine learning (ML) surrogate models are increasingly used in engineering analysis and design to replace computationally expensive simulation models, significantly reducing computational cost and accelerating decision-making processes. However, ML predictions contain inherent errors, often estimated as model uncertainty, which is coupled with variability in model inputs. Accurately quantifying and propagating these combined uncertainties is essential for generating reliable engineering predictions. This paper presents a robust framework based on Polynomial Chaos Expansion (PCE) to handle joint input and model uncertainty propagation. While the approach applies broadly to general ML surrogates, we focus on Gaussian Process regression models, which provide explicit predictive distributions for model uncertainty. By transforming all random inputs into a unified standard space, a PCE surrogate model is constructed, allowing efficient and accurate calculation of the mean and standard deviation of the output. The proposed methodology also offers a mechanism for global sensitivity analysis, enabling the accurate quantification of the individual contributions of input variables and ML model uncertainty to the overall output variability. This approach provides a computationally efficient and interpretable framework for comprehensive uncertainty quantification, supporting trustworthy ML predictions in downstream engineering applications.