  This paper introduces a surrogate modeling scheme based on Grassmannian
manifold learning to be used for cost-efficient predictions of high-dimensional
stochastic systems. The method exploits subspace-structured features of each
solution by projecting it onto a Grassmann manifold. The method utilizes a
solution clustering approach in order to identify regions of the parameter
space over which solutions are sufficiently similarly such that they can be
interpolated on the Grassmannian. In this clustering, the reduced-order
solutions are partitioned into disjoint clusters on the Grassmann manifold
using the eigen-structure of properly defined Grassmannian kernels and, the
Karcher mean of each cluster is estimated. Then, the points in each cluster are
projected onto the tangent space with origin at the corresponding Karcher mean
using the exponential mapping. For each cluster, a Gaussian process regression
model is trained that maps the input parameters of the system to the reduced
solution points of the corresponding cluster projected onto the tangent space.
Using this Gaussian process model, the full-field solution can be efficiently
predicted at any new point in the parameter space. In certain cases, the
solution clusters will span disjoint regions of the parameter space. In such
cases, for each of the solution clusters we utilize a second, density-based
spatial clustering to group their corresponding input parameter points in the
Euclidean space. The proposed method is applied to two numerical examples. The
first is a nonlinear stochastic ordinary differential equation with uncertain
initial conditions. The second involves modeling of plastic deformation in a
model amorphous solid using the Shear Transformation Zone theory of plasticity.
