Data-driven projection pursuit adaptation of polynomial chaos expansions for dependent high-dimensional parameters

Abstract

Uncertainty quantification (UQ) and inference involving a large number of parameters are valuable tools for problems associated with heterogeneous and non-stationary behaviors. The difficulty with these problems is exacerbated when these parameters are statistically dependent requiring statistical characterization over joint measures. Probabilistic modeling methodologies stand as effective tools in the realms of UQ and inference. Among these, polynomial chaos expansions (PCE), when adapted to low-dimensional quantities of interest (QoI), provide effective yet accurate approximations for these QoI in terms of an adapted orthogonal basis. These adaptation techniques have been cast as projection pursuits in Gaussian Hilbert space in what has been referred to as a projection pursuit adaptation (PPA) by Xiaoshu Zeng and Roger Ghanem (2023). The PPA method efficiently identifies an optimal low-dimensional space for representing the QoI and simultaneously evaluates an optimal PCE within that space. The quality of this approximation clearly depends on the size of the training dataset, which is typically a function of the adapted reduced dimension. The complexity of the problem is thus mediated by the complexity of the low-dimensional quantity of interest and not the complexity of the high-dimensional parameter space.

In this paper, our objective is to tackle the challenge of dependent parameters while constructing the PPA, utilizing a generative data-driven framework that requires a fixed number of pre-evaluated (parameter, QoI) pairs. While PCE approaches dealing with dependent input parameters have already been introduced by Christian Soize and Roger Ghanem (2004) their coupling with basis adaptation remains an outstanding task without which they remain plagued by the curse of dimensionality. For modest-sized parameters, mapping such as the Rosenblatt transformation can be employed to decouple the dependent variables. This strategy requires access to the joint distribution of the random variables which is usually lacking, requiring significantly more data than is typically available. To overcome these limitations, we propose leveraging multivariate Regular Vine (R-vine) copulas to encapsulate the dependency structure within parameters, manifested as a joint cumulative density function (CDF). The Rosenblatt transformation can then be applied to decouple the dependent input data, mapping them to samples from independent Gaussian variables. Conversely, we can generate dependent samples from independent Gaussian variables while maintaining the learned dependencies. This generative capability ensures that the reconstructed dependency structure is faithfully preserved in the generated samples. Endowed with the ability to diagonalize measures on product spaces, the R-vine copula blends seamlessly with the PPA method, resulting in a unified procedure for constructing optimally reduced PCE models tailored for high-dimensional problems with dependent parameter spaces. The proposed methodology attains remarkable accuracy for both UQ and inference. In the latter, the constructed PCE model adeptly serves as a generative and convergent surrogate model for machine learning regression. The efficiency of the proposed methodology is validated through two distinct applications: water flow through a borehole and structural dynamics.