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贝叶斯推断近似方法，已在诸多分类任务中成功应用。本研究旨在系统探究该方法在多输出回归任务中预测导弹构型气动性能时的表现。通过分析集成中神经网络数量对预测的影响，观察到估计不确定性存在明显的置信不足趋势。基于此，我们提出应用事后校准方法的深度集成框架，并验证了其改进的不确定性量化性能。该框架与工程中应用最广泛的不确定性量化模型——高斯过程回归进行比较，证明其在回归精度、估计不确定性可靠性及训练效率方面均具有更优表现。最后，分析了所提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会导致完全不同的探索特性。由于本研究未针对具体问题提出特殊假设，该框架可无缝应用于并扩展到任意回归任务。