Data-Driven Polynomial Chaos Expansions for Uncertainty Quantification
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Uncertainties exist in both physics-based and data-driven models of systems. Understanding how system inputs affect a system output's uncertainty is essential to improve system outputs such as quality and productivity. Variance-based sensitivity analysis, which is widely used for uncertainty quantification, characterizes how the output variance is propagated from inputs. To estimate the variance-based sensitivity indices of the output with respect to inputs, polynomial chaos expansions (PCEs) are widely used. However, a majority of existing PCEs impose parametric distributional assumptions on inputs. Furthermore, existing sensitivity indices for dependent inputs impose strong assumptions on the dependence structure of the inputs or lack interpretability. Although recent studies proposed fully data-driven PCEs without strong assumptions on inputs, these PCEs are generally inefficient because the minimally required number of observations increases exponentially in the number of the inputs. To address these challenges, three data-driven PCEs are proposed in this dissertation. We first propose the sparse network PCE (SN-PCE) model for a broad class of systems whose input-output relationships are expressed as directed acyclic graphs. The proposed SN-PCE model accurately estimates variance-based sensitivity indices with far fewer observations than state-of-the-art black-box methods. Next, we propose data-driven sensitivity indices by constructing ordered partitions of linearly independent polynomials of dependent inputs for PCEs. The proposed sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge of the dependence structure of the inputs. Finally, we propose a data-driven algorithm to build sparse PCEs for models with dependent inputs. The proposed algorithm not only reduces the number of minimally required observations but also improves upon the numerical stability and estimation accuracy of a state-of-the-art sparse PCE.