  We are interested in the development of surrogate models for uncertainty
quantification and propagation in problems governed by stochastic PDEs using a
deep convolutional encoder-decoder network in a similar fashion to approaches
considered in deep learning for image-to-image regression tasks. Since normal
neural networks are data intensive and cannot provide predictive uncertainty,
we propose a Bayesian approach to convolutional neural nets. A recently
introduced variational gradient descent algorithm based on Stein's method is
scaled to deep convolutional networks to perform approximate Bayesian inference
on millions of uncertain network parameters. This approach achieves state of
the art performance in terms of predictive accuracy and uncertainty
quantification in comparison to other approaches in Bayesian neural networks as
well as techniques that include Gaussian processes and ensemble methods even
when the training data size is relatively small. To evaluate the performance of
this approach, we consider standard uncertainty quantification benchmark
problems including flow in heterogeneous media defined in terms of limited
data-driven permeability realizations. The performance of the surrogate model
developed is very good even though there is no underlying structure shared
between the input (permeability) and output (flow/pressure) fields as is often
the case in the image-to-image regression models used in computer vision
problems. Studies are performed with an underlying stochastic input
dimensionality up to $4,225$ where most other uncertainty quantification
methods fail. Uncertainty propagation tasks are considered and the predictive
output Bayesian statistics are compared to those obtained with Monte Carlo
estimates.
