Efficient Development and Application of Taylor Series Expansions as Surrogate Models for Uncertainty Quantification

A derivative-based Uncertainty Quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. HYPercomplex Automatic Differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities were used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices . Algebraic expressions of these central moments and Sobol' indices of the Taylor series expansion were derived for any distribution of the random variables. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

Increasing the expansion order of the Taylor series can improve the accuracy; however, the number of higher-order derivatives needed to be computed grows rapidly. In addition, not all terms will significantly contribute to the accuracy of the surrogate model. For these reasons, methods to construct a sparse Taylor series expansion have been developed which identify the important derivatives in the expansion needed to obtain a sufficient approximation of the variance of the output of the high-fidelity model. Three methods are presented to identify important derivatives in the Taylor series. The first method computes the weight corresponding to each derivative in the algebraic expression of the variance of the Taylor series expansion, which is done prior to derivative computations. The second method uses Sobol' indices of lower-order Taylor series expansions to predict important derivatives in the higher-order expansions. The third method uses regression-based feature selection methods to predict important terms in the Taylor series using a small training set of high-fidelity model evaluations. These sparse Taylor series expansion methodologies were shown to sufficiently approximate the variance of the high-fidelity model with significantly less computational cost compared to the complete Taylor series expansion linear regression models, and polynomial chaos expansion surrogate models in a numerical example problem.