Data-driven dimensionally decomposed generalized polynomial chaos expansion for forward uncertainty quantification

TL;DR

This work tackles forward uncertainty quantification for high-dimensional inputs with unknown distributions by developing a data-driven DD-GPCE framework. It estimates input densities directly from samples using KDE and applies a smoothed bootstrap to reduce estimator bias, then constructs measure-consistent orthonormal polynomials via a whitening transformation to form a DD-GPCE surrogate with low-order interactions. Coefficients are estimated through standard least squares, enabling accurate propagation of uncertainty without distributional assumptions. Numerical results on mathematical functions, a ten-bar truss, and an aircraft structure demonstrate that the data-driven approach yields superior mean/variance estimates compared with Gaussian-assumed data-driven methods, while retaining scalability through low-order interaction modeling. Overall, the method broadens DD-GPCE applicability to practical UQ tasks with limited data and unknown input distributions.

Abstract

Dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) efficiently performs forward uncertainty quantification (UQ) in complex engineering systems with high-dimensional random inputs of arbitrary distributions. However, constructing the measure-consistent orthonormal polynomial bases in DD-GPCE requires prior knowledge of input distributions, which is often unavailable in practice. This work introduces a data-driven DD-GPCE method that eliminates the need for such prior knowledge, extending its applicability to UQ with high-dimensional inputs. Input distributions are inferred directly from sample data using smoothed-bootstrap kernel density estimation (KDE), while the DD-GPCE framework enables KDE to handle high-dimensional inputs through low-dimensional marginal estimation. We then use the estimated input distributions to perform a whitening transformation via Monte Carlo Simulation, which enables generation of measure-consistent orthonormal basis functions. We demonstrate the accuracy of the proposed method in both mathematical examples and stochastic dynamic analysis for a practical three-dimensional mobility design involving twenty random inputs. The results indicate that the proposed method produces more accurate estimates of the output mean and variance compared to the conventional data-driven approach that assumes Gaussian input distributions.

Figures (26)

• Figure 1: Contour plots of probability density estimated via KDE with using the original samples (center) and the smoothed bootstrap samples (right) compared to the true density contour plot (left).
• Figure 2: A computation flow for generating the data-driven DD-GPCE approximation
• Figure 4: A ten-bar truss structure (Example 2).
• Figure 7: The DA50 RG model (Example 3) by Diamond Aircraft is made of composite material, featuring a wingspan of 13.41m, a height of 2.95m, and a length of 9.24m. The geometric model was designed with CATIA V5. The left bottom figure shows the modal analysis result conducted using the parametric model with 11 design variables and 9 material properties.
• Figure 8: Comparison of KDE-based distribution estimation for $X_{14}$. The left shows results without bandwidth scaling, while the right applies scaling by the sample standard deviation, yielding a closer match to the original distribution.