  The present work addresses the issue of accurate stochastic approximations in
high-dimensional parametric space using tools from uncertainty quantification
(UQ). The basis adaptation method and its accelerated algorithm in polynomial
chaos expansions (PCE) were recently proposed to construct low-dimensional
approximations adapted to specific quantities of interest (QoI). The present
paper addresses one difficulty with these adaptations, namely their reliance on
quadrature point sampling, which limits the reusability of potentially
expensive samples. Projection pursuit (PP) is a statistical tool to find the
``interesting'' projections in high-dimensional data and thus bypass the
curse-of-dimensionality. In the present work, we combine the fundamental ideas
of basis adaptation and projection pursuit regression (PPR) to propose a novel
method to simultaneously learn the optimal low-dimensional spaces and PCE
representation from given data. While this projection pursuit adaptation (PPA)
can be entirely data-driven, the constructed approximation exhibits mean-square
convergence to the solution of an underlying governing equation and is thus
subject to the same physics constraints. The proposed approach is demonstrated
on a borehole problem and a structural dynamics problem, demonstrating the
versatility of the method and its ability to discover low-dimensional manifolds
with high accuracy with limited data. In addition, the method can learn
surrogate models for different quantities of interest while reusing the same
data set.
