Polynomial Chaos Expansions (PCEs) are widely recognized for their efficient computational performance in surrogate modeling. Yet, a robust framework to quantify local model errors is still lacking. While the local uncertainty of PCE prediction can be captured using bootstrap resampling, other methods offering more rigorous statistical guarantees are needed, especially in the context of small training datasets. Recently, conformal predictions have demonstrated strong potential in machine learning, providing statistically robust and model-agnostic prediction intervals. Due to its generality and versatility, conformal prediction is especially valuable, as it can be adapted to suit a variety of problems, making it a compelling choice for PCE-based surrogate models. In this contribution, we explore its application to PCE-based surrogate models. More precisely, we present the integration of two conformal prediction methods, namely the full conformal and the Jackknife+ approaches, into both full and sparse PCEs. For full PCEs, we introduce computational shortcuts inspired by the inherent structure of regression methods to optimize the implementation of both conformal methods. For sparse PCEs, we incorporate the two approaches with appropriate modifications to the inference strategy, thereby circumventing the non-symmetrical nature of the regression algorithm and ensuring valid prediction intervals. Our developments yield better-calibrated prediction intervals for both full and sparse PCEs, achieving superior coverage over existing approaches, such as the bootstrap, while maintaining a moderate computational cost.

翻译：多项式混沌展开（PCEs）因其在代理建模中高效的计算性能而广受认可。然而，目前仍缺乏一个量化局部模型误差的稳健框架。虽然PCE预测的局部不确定性可通过自助重采样方法捕捉，但仍需要其他能提供更严格统计保证的方法，尤其是在小规模训练数据集的背景下。近年来，保形预测在机器学习领域展现出巨大潜力，能够提供统计上稳健且与模型无关的预测区间。由于其普适性和多功能性，保形预测尤其具有价值，可适配各类问题，这使其成为基于PCE的代理模型的一个引人注目的选择。在本文中，我们探讨了其在基于PCE的代理模型中的应用。具体而言，我们介绍了两种保形预测方法——完全保形法与Jackknife+法——在完全与稀疏PCE中的整合。对于完全PCE，我们借鉴回归方法的内在结构，引入计算捷径以优化两种保形方法的实现。对于稀疏PCE，我们将这两种方法结合到推断策略中并作适当调整，从而规避回归算法的非对称特性，确保有效的预测区间。我们的研究为完全与稀疏PCE均提供了校准更优的预测区间，在保持适中计算成本的同时，相较于自助法等方法实现了更优越的覆盖性能。