  In this work we introduce a manifold learning-based method for uncertainty
quantification (UQ) in systems describing complex spatiotemporal processes. Our
first objective is to identify the embedding of a set of high-dimensional data
representing quantities of interest of the computational or analytical model.
For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear
dimension reduction technique which allows us to reduce the dimensionality of
the data and identify meaningful geometric descriptions in a parsimonious and
inexpensive manner. Polynomial chaos expansion is then used to construct a
mapping between the stochastic input parameters and the diffusion coordinates
of the reduced space. An adaptive clustering technique is proposed to identify
an optimal number of clusters of points in the latent space. The similarity of
points allows us to construct a number of geometric harmonic emulators which
are finally utilized as a set of inexpensive pre-trained models to perform an
inverse map of realizations of latent features to the ambient space and thus
perform accurate out-of-sample predictions. Thus, the proposed method acts as
an encoder-decoder system which is able to automatically handle very
high-dimensional data while simultaneously operating successfully in the
small-data regime. The method is demonstrated on two benchmark problems and on
a system of advection-diffusion-reaction equations which model a first-order
chemical reaction between two species. In all test cases, the proposed method
is able to achieve highly accurate approximations which ultimately lead to the
significant acceleration of UQ tasks.
