Machine learning methods for the construction of data-driven reduced order model models are used in an increasing variety of engineering domains, especially as a supplement to expensive computational fluid dynamics for design problems. An important check on the reliability of surrogate models is Uncertainty Quantification (UQ), a self assessed estimate of the model error. Accurate UQ allows for cost savings by reducing both the required size of training data sets and the required safety factors, while poor UQ prevents users from confidently relying on model predictions. We examine several machine learning techniques, including both Gaussian processes and a family UQ-augmented neural networks: Ensemble neural networks (ENN), Bayesian neural networks (BNN), Dropout neural networks (D-NN), and Gaussian neural networks (G-NN). We evaluate UQ accuracy (distinct from model accuracy) using two metrics: the distribution of normalized residuals on validation data, and the distribution of estimated uncertainties. We apply these metrics to two model data sets, representative of complex dynamical systems: an ocean engineering problem in which a ship traverses irregular wave episodes, and a dispersive wave turbulence system with extreme events, the Majda-McLaughlin-Tabak model. We present conclusions concerning model architecture and hyperparameter tuning.

翻译：用于构建数据驱动降阶模型的机器学习方法正越来越多地被应用于各类工程领域，特别是在设计问题中作为昂贵计算流体动力学的补充手段。代理模型可靠性的重要检验指标是不确定性量化（UQ），即模型误差的自我评估估计。准确的UQ既能降低所需训练数据集规模，又能减小安全系数，从而节约成本；而低质量的UQ则使用户无法自信地依赖模型预测。我们研究了多种机器学习技术，包括高斯过程以及一系列UQ增强型神经网络：集成神经网络（ENN）、贝叶斯神经网络（BNN）、丢弃神经网络（D-NN）和高斯神经网络（G-NN）。我们采用两个指标评估UQ精度（与模型精度相区分）：验证数据上归一化残差的分布，以及估计不确定性的分布。我们将这些指标应用于两个代表复杂动力系统的模型数据集：一个海洋工程问题（船舶穿越不规则波浪过程），以及一个存在极端事件的色散波湍流系统——Majda-McLaughlin-Tabak模型。最后，我们给出了关于模型架构与超参数调优的结论。