Uncertainty Quantification in Three Dimensional Natural Convection using Polynomial Chaos Expansion and Deep Neural Networks
For engineers performing uncertainty quantification in fluid dynamics, this work offers a computationally cheaper surrogate modeling alternative to PCE for high-dimensional stochastic problems.
This paper analyzes uncertainty propagation in 3D natural convection using polynomial chaos expansion (PCE) for low-dimensional stochastic inputs and deep neural networks (DNNs) for high-dimensional boundary temperature uncertainty. The DNN surrogate achieves reasonable accuracy with fewer simulations than PCE, reducing computational cost for high stochastic dimensions (e.g., 32).
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.