  Fine-scale simulation of complex systems governed by multiscale partial
differential equations (PDEs) is computationally expensive and various
multiscale methods have been developed for addressing such problems. In
addition, it is challenging to develop accurate surrogate and uncertainty
quantification models for high-dimensional problems governed by stochastic
multiscale PDEs using limited training data. In this work to address these
challenges, we introduce a novel hybrid deep-learning and multiscale approach
for stochastic multiscale PDEs with limited training data. For demonstration
purposes, we focus on a porous media flow problem. We use an image-to-image
supervised deep learning model to learn the mapping between the input
permeability field and the multiscale basis functions. We introduce a Bayesian
approach to this hybrid framework to allow us to perform uncertainty
quantification and propagation tasks. The performance of this hybrid approach
is evaluated with varying intrinsic dimensionality of the permeability field.
Numerical results indicate that the hybrid network can efficiently predict well
for high-dimensional inputs.
