  We propose a method for quantifying uncertainty in high-dimensional PDE
systems with random parameters, where the number of solution evaluations is
small. Parametric PDE solutions are often approximated using a spectral
decomposition based on polynomial chaos expansions. For the class of systems we
consider (i.e., high dimensional with limited solution evaluations) the
coefficients are given by an underdetermined linear system in a regression
formulation. This implies additional assumptions, such as sparsity of the
coefficient vector, are needed to approximate the solution. Here, we present an
approach where we assume the coefficients are close to the range of a
generative model that maps from a low to a high dimensional space of
coefficients. Our approach is inspired be recent work examining how generative
models can be used for compressed sensing in systems with random Gaussian
measurement matrices. Using results from PDE theory on coefficient decay rates,
we construct an explicit generative model that predicts the polynomial chaos
coefficient magnitudes. The algorithm we developed to find the coefficients,
which we call GenMod, is composed of two main steps. First, we predict the
coefficient signs using Orthogonal Matching Pursuit. Then, we assume the
coefficients are within a sparse deviation from the range of a sign-adjusted
generative model. This allows us to find the coefficients by solving a
nonconvex optimization problem, over the input space of the generative model
and the space of sparse vectors. We obtain theoretical recovery results for a
Lipschitz continuous generative model and for a more specific generative model,
based on coefficient decay rate bounds. We examine three high-dimensional
problems and show that, for all three examples, the generative model approach
outperforms sparsity promoting methods at small sample sizes.
