  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.
