Bounds optimization of model response moments: a twin-engine Bayesian active learning method

The efficient propagation of imprecise probabilities through expensive simulators has emerged to be one of the great challenges for mixed uncertainty quantification in computational mechanics. An active learning method, named Collaborative and Adaptive Bayesian Optimization (CABO), is developed for tackling this challenge by combining Bayesian Probabilistic Optimization and Bayesian Probabilistic Integration. Two learning functions are introduced as engines for CABO, where one is introduced for realizing the adaptive optimization search in the epistemic uncertainty space, and the other one is developed for adaptive integration in the aleatory uncertainty space. These two engines work in a collaborative way to create optimal design points adaptively in the joint uncertainty space, by which a Gaussian process regression model is trained and updated to approach the bounds of model response moments with pre-specified error tolerances. The effectiveness of CABO is demonstrated using a numerical example and two engineering benchmarks.