This work presents novel extensions for combining two frameworks for quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainties in the modeling of engineered systems. The data-consistent (DC) framework poses an inverse problem and solution for quantifying aleatoric uncertainties in terms of pullback and push-forward measures for a given Quantity of Interest (QoI) map. Unfortunately, a pre-specified QoI map is not always available a priori to the collection of data associated with system outputs. The data themselves are often polluted with measurement errors (i.e., epistemic uncertainties), which complicates the process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ) framework defines a formal three-step machine-learning enabled process for transforming noisy datasets into samples of a learned QoI map to enable DC-based inversion. We develop a robust filtering step in LUQ that can learn the most useful quantitative information present in spatio-temporal datasets. The learned QoI map transforms simulated and observed datasets into distributions to perform DC-based inversion. We also develop a DC-based inversion scheme that iterates over time as new spatial datasets are obtained and utilizes quantitative diagnostics to identify both the quality and impact of inversion at each iteration. Reproducing Kernel Hilbert Space theory is leveraged to mathematically analyze the learned QoI map and develop a quantitative sufficiency test for evaluating the filtered data. An illustrative example is utilized throughout while the final two examples involve the manufacturing of shells of revolution to demonstrate various aspects of the presented frameworks.

翻译：本文提出新型拓展方法，用于融合两种框架以量化工程系统建模中偶然不确定性（即不可约性）与认知不确定性（即可约性）的双重来源。数据一致性（DC）框架通过给定兴趣量（QoI）映射的拉回测度与推前测度构建反问题及求解方案以量化偶然不确定性。然而，预定义的QoI映射并非总能在收集系统输出数据前获得，而数据本身常受测量误差（即认知不确定性）污染，进一步增加了指定有效QoI的难度。学习不确定量（LUQ）框架定义了规范的三步机器学习流程，可将含噪声数据集转化为学习型QoI映射的样本，从而支持基于DC的反演。我们开发了LUQ中的稳健滤波步骤，能够学习时空数据集中最具价值的定量信息。学习到的QoI映射将仿真与观测数据集转化为分布，以执行基于DC的反演。我们还提出一种随时间迭代的DC反演方案，在获取新空间数据集时利用定量诊断工具识别每次迭代的反演质量与影响。借助再生核希尔伯特空间理论对学习型QoI映射进行数学分析，并开发了滤波数据定量充分性检验方法。全文以说明性例题贯穿始终，最后两个例题涉及回转壳制造，以展示所提框架的多个应用特性。