Machine learning-based reliability analysis methods have shown great advancements for their computational efficiency and accuracy. Recently, many efficient learning strategies have been proposed to enhance the computational performance. However, few of them explores the theoretical optimal learning strategy. In this article, we propose several theorems that facilitates such exploration. Specifically, cases that considering and neglecting the correlations among the candidate design samples are well elaborated. Moreover, we prove that the well-known U learning function can be reformulated to the optimal learning function for the case neglecting the Kriging correlation. In addition, the theoretical optimal learning strategy for sequential multiple training samples enrichment is also mathematically explored through the Bayesian estimate with the corresponding lost functions. Simulation results show that the optimal learning strategy considering the Kriging correlation works better than that neglecting the Kriging correlation and other state-of-the art learning functions from the literatures in terms of the reduction of number of evaluations of performance function. However, the implementation needs to investigate very large computational resource.

翻译：基于机器学习的可靠性分析方法因其计算效率和准确性而取得了显著进展。近年来，许多高效的学习策略被提出以提升计算性能，然而鲜有研究探讨理论最优学习策略。本文提出若干定理以推动该探索，具体而言，系统阐述了考虑与忽略候选设计样本间相关性的两类情形。此外，我们证明了著名的U学习函数可重新表述为忽略克里金相关性情形下的最优学习函数。同时，通过贝叶斯估计与相应损失函数，从数学角度探讨了序贯多训练样本扩充的理论最优学习策略。仿真结果表明，考虑克里金相关性的最优学习策略在减少性能函数评估次数方面，优于忽略该相关性的策略及文献中的其他先进学习函数，但该策略的实现需消耗极大的计算资源。