Bayesian optimized deep ensemble for uncertainty quantification of deep neural networks: a system safety case study on sodium fast reactor thermal stratification modeling

Accurate predictions and uncertainty quantification (UQ) are essential for decision-making in risk-sensitive fields such as system safety modeling. Deep ensembles (DEs) are efficient and scalable methods for UQ in Deep Neural Networks (DNNs); however, their performance is limited when constructed by simply retraining the same DNN multiple times with randomly sampled initializations. To overcome this limitation, we propose a novel method that combines Bayesian optimization (BO) with DE, referred to as BODE, to enhance both predictive accuracy and UQ. We apply BODE to a case study involving a Densely connected Convolutional Neural Network (DCNN) trained on computational fluid dynamics (CFD) data to predict eddy viscosity in sodium fast reactor thermal stratification modeling. Compared to a manually tuned baseline ensemble, BODE estimates total uncertainty approximately four times lower in a noise-free environment, primarily due to the baseline's overestimation of aleatoric uncertainty. Specifically, BODE estimates aleatoric uncertainty close to zero, while aleatoric uncertainty dominates the total uncertainty in the baseline ensemble. We also observe a reduction of more than 30% in epistemic uncertainty. When Gaussian noise with standard deviations of 5% and 10% is introduced into the data, BODE accurately fits the data and estimates uncertainty that aligns with the data noise. These results demonstrate that BODE effectively reduces uncertainty and enhances predictions in data-driven models, making it a flexible approach for various applications requiring accurate predictions and robust UQ.