  This paper analyzes the effects of input uncertainties on the outputs of a
three dimensional natural convection problem in a differentially heated cubical
enclosure. Two different cases are considered for parameter uncertainty
propagation and global sensitivity analysis. In case A, stochastic variation is
introduced in the two non-dimensional parameters (Rayleigh and Prandtl numbers)
with an assumption that the boundary temperature is uniform. Being a two
dimensional stochastic problem, the polynomial chaos expansion (PCE) method is
used as a surrogate model. Case B deals with non-uniform stochasticity in the
boundary temperature. Instead of the traditional Gaussian process model with
the Karhunen-Lo$\grave{e}$ve expansion, a novel approach is successfully
implemented to model uncertainty in the boundary condition. The boundary is
divided into multiple domains and the temperature imposed on each domain is
assumed to be an independent and identically distributed (i.i.d) random
variable. Deep neural networks are trained with the boundary temperatures as
inputs and Nusselt number, internal temperature or velocities as outputs. The
number of domains which is essentially the stochastic dimension is 4, 8, 16 or
32. Rigorous training and testing process shows that the neural network is able
to approximate the outputs to a reasonable accuracy. For a high stochastic
dimension such as 32, it is computationally expensive to fit the PCE. This
paper demonstrates a novel way of using the deep neural network as a surrogate
modeling method for uncertainty quantification with the number of simulations
much fewer than that required for fitting the PCE, thus, saving the
computational cost.
