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.

翻译：在系统安全建模等风险敏感领域中，准确的预测与不确定性量化对决策至关重要。深度集成是深度神经网络中进行不确定性量化的一种高效且可扩展的方法；然而，当仅通过随机采样初始化多次重训练同一深度神经网络来构建时，其性能会受到限制。为克服此限制，我们提出一种将贝叶斯优化与深度集成相结合的新方法，称为BODE，以同时提升预测准确性与不确定性量化效果。我们将BODE应用于一个案例研究，该研究使用在计算流体动力学数据上训练的密集连接卷积神经网络来预测钠冷快堆热分层建模中的涡粘性。与手动调参的基线集成相比，在无噪声环境中，BODE估计的总不确定性降低了约四倍，这主要归因于基线方法高估了偶然不确定性。具体而言，BODE估计的偶然不确定性接近于零，而基线集成中的总不确定性主要由偶然不确定性主导。我们还观察到认知不确定性降低了30%以上。当向数据中引入标准差为5%和10%的高斯噪声时，BODE能准确拟合数据并估计出与数据噪声一致的不确定性。这些结果表明，BODE能有效降低数据驱动模型中的不确定性并提升预测性能，使其成为需要准确预测和鲁棒不确定性量化的各种应用的一种灵活方法。