An expected integrated error reduction function for accelerating Bayesian active learning of failure probability

The combination of active learning with surrogate model (e.g., Gaussian Process Regression, GPR) for structural reliability analysis has been extensively studied and proved to be of superiority due to the high efficiency. The performance of an active learning algorithm is determined by the utilized acquisition function, to a great extent, as it dominates the generation of the training data. The traditional acquisition functions usually fail to incorporate the spatial correlation information revealed by the surrogate model, thus still have plenty of room for improvement. We propose a new acquisition function, named as Expected Integrated Error Reduction (EIER) function, for active learning of the failure probability with a smaller number of simulator calls. Mathematically, the value of the EIER function at an unobserved point measures the expected integrated reduction of the probability of mis-classifying all points over the full input space, once this point is added to train the GPR model. The numerical error of the failure probability is evaluated with the posterior confidence intervals and/or coefficient of variation, numerically computed with an efficient sampling strategy, and is then served as the stopping criteria. The superiority of the proposed improvements is demonstrated with a set of numerical and engineering examples.