Deep ensemble is a simple and straightforward approach for approximating Bayesian inference and has been successfully applied to many classification tasks. This study aims to comprehensively investigate this approach in the multi-output regression task to predict the aerodynamic performance of a missile configuration. By scrutinizing the effect of the number of neural networks used in the ensemble, an obvious trend toward underconfidence in estimated uncertainty is observed. In this context, we propose the deep ensemble framework that applies the post-hoc calibration method, and its improved uncertainty quantification performance is demonstrated. It is compared with Gaussian process regression, the most prevalent model for uncertainty quantification in engineering, and is proven to have superior performance in terms of regression accuracy, reliability of estimated uncertainty, and training efficiency. Finally, the impact of the suggested framework on the results of Bayesian optimization is examined, showing that whether or not the deep ensemble is calibrated can result in completely different exploration characteristics. This framework can be seamlessly applied and extended to any regression task, as no special assumptions have been made for the specific problem used in this study.

翻译：深度集成是一种简单直接的近似贝叶斯推断方法，已成功应用于诸多分类任务。本研究旨在系统探究该方法在多输出回归任务中的应用，以预测导弹构型的气动性能。通过细致分析集成中神经网络数量对结果的影响，观察到估计不确定性存在明显的欠自信趋势。为此，我们提出一种集成后校准的深度集成框架，并证明了其在不确定性量化性能上的改进。与工程中不确定性量化最常用的高斯过程回归模型相比，该框架在回归精度、估计不确定性的可靠性以及训练效率方面均展现出更优性能。最后，本研究考察了所提议框架对贝叶斯优化结果的影响，表明深度集成是否经过校准可能导致完全不同的探索特性。由于未对研究中使用的特定问题做出特殊假设，该框架可无缝应用于并推广至任何回归任务。