  We develop a novel deep learning method for uncertainty quantification in
stochastic partial differential equations based on Bayesian neural network
(BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior
distribution of the parameters in deep neural networks by performing Bayesian
inference on the network parameters. The posterior distribution is efficiently
sampled using HMC to quantify uncertainties in the system. Several numerical
examples are shown for both forward and inverse problems in high dimension to
demonstrate the effectiveness of the proposed method for uncertainty
quantification. These also show promising results that the computational cost
is almost independent of the dimension of the problem demonstrating the
potential of the method for tackling the so-called curse of dimensionality.
