Towards Quantifying Calibrated Uncertainty via Deep Ensembles in Multi-output Regression Task

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.