Bayesian³ Active Learning for Regularized Arbitrary Multi-Element Polynomial Chaos Using Information Theory

Machine learning, surrogate modeling, and uncertainty quantification pose challenges in data-poor applications that arise due to limited availability of measurement data or with computationally expensive models. Specialized models, derived from Gaussian process emulators (GPEs) or polynomial chaos expansions (PCEs), are often used when only limited numbers of training points are available. The PCE (or its data-driven version, the arbitrary polynomial chaos) is based on a global representation informed by the distributions of model parameters, whereas GPEs rely on a local kernel and additionally assess the uncertainty of the surrogate itself. Oscillation-mitigating localizations of the PCE result in increased degrees of freedom (DoF), requiring more training samples. As applications such as Bayesian inference (BI) require highly accurate surrogates, even specialized models like PCE or GPE require a substantial amount of training data. Bayesian3 active learning (B3AL) on GPEs, based on information theory (IT), can reduce the necessary number of training samples for BI. IT-based ideas for B3AL are not yet directly transferable to the PCE family, as this family lacks awareness of surrogate uncertainty by design. In the present work, we introduce a Bayesian regularized version of localized arbitrary polynomial chaos to build surrogate models. Equipped with Gaussian emulator properties, our fully adaptive framework is enhanced with B3AL methods designed to achieve reliable surrogate models for BI while efficiently selecting training samples via IT. The effectiveness of the proposed methodology is demonstrated by comprehensive evaluations on several numerical examples.