A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness

TL;DR

This paper introduces a novel data-driven framework for constructing sparse surrogate models that can handle arbitrary, mutually dependent randomness in high-dimensional uncertainty quantification problems, overcoming limitations of traditional methods.

Contribution

The paper develops a new approach to construct orthonormal polynomial bases for arbitrary dependent distributions, enabling accurate sparse surrogate modeling without assuming independence.

Findings

Effective in high-dimensional PDE problems

Accurately models systems with implicit non-Gaussian measures

Maintains orthogonality after rotation for dependent variables

Abstract

The challenge of quantifying uncertainty propagation in real-world systems is rooted in the high-dimensionality of the stochastic input and the frequent lack of explicit knowledge of its probability distribution. Traditional approaches show limitations for such problems. To address these difficulties, we have developed a general framework of constructing surrogate models on spaces of stochastic input with arbitrary probability measure irrespective of the mutual dependencies between individual components and the analytical form. The present Data-driven Sparsity-enhancing Rotation for Arbitrary Randomness (DSRAR) framework includes a data-driven construction of multivariate polynomial basis for arbitrary mutually dependent probability measure and a sparsity enhancement rotation procedure. This sparsity-enhancing rotation method was initially proposed in our previous work [1] for Gaussian distributions, which may not be feasible for non-Gaussian distributions due to the loss of orthogonality after the rotation. To remedy such difficulties, we developed the new approach to construct orthonormal polynomials for arbitrary mutually dependent (amdP) randomness, ensuring the constructed basis maintains the orthogonality with respect to the density of the rotated random vector, where directly applying the regular polynomial chaos including arbitrary polynomial chaos (aPC) [2] shows limitations due to the assumption of the mutual independence between the components of the random inputs. The developed DSRAR framework leads to accurate recovery of a sparse representation of the target functions. The effectiveness of our method is demonstrated in challenging problems such as PDEs and realistic molecular systems where the underlying density is implicitly represented by a large collection of sample data, as well as systems with explicitly given non-Gaussian probabilistic measures.