  Stochastic partial differential equations (SPDEs) are ubiquitous in
engineering and computational sciences. The stochasticity arises as a
consequence of uncertainty in input parameters, constitutive relations,
initial/boundary conditions, etc. Because of these functional uncertainties,
the stochastic parameter space is often high-dimensional, requiring hundreds,
or even thousands, of parameters to describe it. This poses an insurmountable
challenge to response surface modeling since the number of forward model
evaluations needed to construct an accurate surrogate grows exponentially with
the dimension of the uncertain parameter space; a phenomenon referred to as the
\textit{curse of dimensionality}. State-of-the-art methods for high-dimensional
uncertainty propagation seek to alleviate the curse of dimensionality by
performing dimensionality reduction in the uncertain parameter space. However,
one still needs to perform forward model evaluations that potentially carry a
very high computational burden. We propose a novel methodology for
high-dimensional uncertainty propagation of elliptic SPDEs which lifts the
requirement for a deterministic forward solver. Our approach is as follows. We
parameterize the solution of the elliptic SPDE using a deep residual network
(ResNet). In a departure from the traditional squared residual (SR) based loss
function for training the ResNet, we introduce a novel physics-informed loss
function derived from variational principles. Specifically, our loss function
is the expectation of the energy functional of the PDE over the stochastic
variables. We demonstrate our solver-free approach through various examples
where the elliptic SPDE is subjected to different types of high-dimensional
input uncertainties. Also, we solve high-dimensional uncertainty propagation
and inverse problems.
