  Constructing surrogate models for uncertainty quantification (UQ) on complex
partial differential equations (PDEs) having inherently high-dimensional
$\mathcal{O}(10^{\ge 2})$ stochastic inputs (e.g., forcing terms, boundary
conditions, initial conditions) poses tremendous challenges. The curse of
dimensionality can be addressed with suitable unsupervised learning techniques
used as a pre-processing tool to encode inputs onto lower-dimensional subspaces
while retaining its structural information and meaningful properties. In this
work, we review and investigate thirteen dimension reduction methods including
linear and nonlinear, spectral, blind source separation, convex and non-convex
methods and utilize the resulting embeddings to construct a mapping to
quantities of interest via polynomial chaos expansions (PCE). We refer to the
general proposed approach as manifold PCE (m-PCE), where manifold corresponds
to the latent space resulting from any of the studied dimension reduction
methods. To investigate the capabilities and limitations of these methods we
conduct numerical tests for three physics-based systems (treated as
black-boxes) having high-dimensional stochastic inputs of varying complexity
modeled as both Gaussian and non-Gaussian random fields to investigate the
effect of the intrinsic dimensionality of input data. We demonstrate both the
advantages and limitations of the unsupervised learning methods and we conclude
that a suitable m-PCE model provides a cost-effective approach compared to
alternative algorithms proposed in the literature, including recently proposed
expensive deep neural network-based surrogates and can be readily applied for
high-dimensional UQ in stochastic PDEs.
