  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.
