Failure probability estimation problem is an crucial task in engineering. In this work we consider this problem in the situation that the underlying computer models are extremely expensive, which often arises in the practice, and in this setting, reducing the calls of computer model is of essential importance. We formulate the problem of estimating the failure probability with expensive computer models as an sequential experimental design for the limit state (i.e., the failure boundary) and propose a series of efficient adaptive design criteria to solve the design of experiment (DOE). In particular, the proposed method employs the deep neural network (DNN) as the surrogate of limit state function for efficiently reducing the calls of expensive computer experiment. A map from the Gaussian distribution to the posterior approximation of the limit state is learned by the normalizing flows for the ease of experimental design. Three normalizing-flows-based design criteria are proposed in this work for deciding the design locations based on the different assumption of generalization error. The accuracy and performance of the proposed method is demonstrated by both theory and practical examples.

翻译：失效概率估计问题是工程领域的一项重要任务。本文考虑实际中常出现的底层计算机模型极为昂贵的情况，在此设定下，减少计算机模型的调用次数至关重要。我们将昂贵计算机模型下的失效概率估计问题表述为针对极限状态（即失效边界）的序贯实验设计，并提出了一系列高效的自适应设计准则以解决实验设计问题。具体而言，所提方法采用深度神经网络作为极限状态函数的代理模型，从而有效减少昂贵计算机实验的调用次数。通过归一化流学习从高斯分布到极限状态后验近似分布的映射，以便于实验设计。本文基于泛化误差的不同假设，提出了三种基于归一化流的设计准则来确定设计点位置。通过理论分析与实际案例验证了所提方法的准确性与有效性。