  State-of-the-art computer codes for simulating real physical systems are
often characterized by a vast number of input parameters. Performing
uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost
always infeasible because of the need to perform hundreds of thousands or even
millions of forward model evaluations in order to obtain convergent statistics.
One, thus, tries to construct a cheap-to-evaluate surrogate model to replace
the forward model solver. For systems with large numbers of input parameters,
one has to deal with the curse of dimensionality - the exponential increase in
the volume of the input space, as the number of parameters increases linearly.
In this work, we demonstrate the use of deep neural networks (DNN) to construct
surrogate models for numerical simulators. We parameterize the structure of the
DNN in a manner that lends the DNN surrogate the interpretation of recovering a
low dimensional nonlinear manifold. The model response is a parameterized
nonlinear function of the low dimensional projections of the input. We think of
this low dimensional manifold as a nonlinear generalization of the notion of
the active subspace. Our approach is demonstrated with a problem on uncertainty
propagation in a stochastic elliptic partial differential equation (SPDE) with
uncertain diffusion coefficient. We deviate from traditional formulations of
the SPDE problem by not imposing a specific covariance structure on the random
diffusion coefficient. Instead, we attempt to solve a more challenging problem
of learning a map between an arbitrary snapshot of the diffusion field and the
response.
