This paper proposes an active learning-based Gaussian process (AL-GP) metamodelling method to estimate the cumulative as well as complementary cumulative distribution function (CDF/CCDF) for forward uncertainty quantification (UQ) problems. Within the field of UQ, previous studies focused on developing AL-GP approaches for reliability (rare event probability) analysis of expensive black-box solvers. A naive iteration of these algorithms with respect to different CDF/CCDF threshold values would yield a discretized CDF/CCDF. However, this approach inevitably leads to a trade-off between accuracy and computational efficiency since both depend (in opposite way) on the selected discretization. In this study, a specialized error measure and a learning function are developed such that the resulting AL-GP method is able to efficiently estimate the CDF/CCDF for a specified range of interest without an explicit dependency on discretization. Particularly, the proposed AL-GP method is able to simultaneously provide accurate CDF and CCDF estimation in their median-low probability regions. Three numerical examples are introduced to test and verify the proposed method.

中文翻译：

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一种新的基于主动学习的高斯过程元建模策略，用于估计前向 UQ 分析中的全概率分布

本文提出了一种基于主动学习的高斯过程 (AL-GP) 元建模方法，用于估计前向不确定性量化 (UQ) 问题的累积和互补累积分布函数 (CDF/CCDF)。在昆士兰大学领域，之前的研究侧重于开发用于对昂贵的黑盒求解器进行可靠性（罕见事件概率）分析的 AL-GP 方法。这些算法相对于不同的 CDF/CCDF 阈值的简单迭代将产生离散化的 CDF/CCDF。然而，这种方法不可避免地导致精度和计算效率之间的权衡，因为两者都依赖于（以相反的方式）所选的离散化。在这项研究中，开发了专门的误差度量和学习函数，使得最终的 AL-GP 方法能够有效地估计指定感兴趣范围的 CDF/CCDF，而无需明确依赖于离散化。特别是，所提出的 AL-GP 方法能够在其中值低概率区域中同时提供准确的 CDF 和 CCDF 估计。引入了三个数值例子来测试和验证所提出的方法。