Two of the most significant challenges in uncertainty propagation pertain to the high computational cost for the simulation of complex physical models and the high dimension of the random inputs. In applications of practical interest both of these problems are encountered and standard methods for uncertainty quantification either fail or are not feasible. To overcome the current limitations, we propose a probabilistic multi-fidelity framework that can exploit lower-fidelity model versions of the original problem in a small data regime. The approach circumvents the curse of dimensionality by learning dependencies between the outputs of high-fidelity models and lower-fidelity models instead of explicitly accounting for the high-dimensional inputs. We complement the information provided by a low-fidelity model with a low-dimensional set of informative features of the stochastic input, which are discovered by employing a combination of supervised and unsupervised dimensionality reduction techniques. The goal of our analysis is an efficient and accurate estimation of the full probabilistic response for a high-fidelity model. Despite the incomplete and noisy information that low-fidelity predictors provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained in the small data regime, i.e., with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty propagation. We illustrate our approach by applying it to challenging numerical examples such as Navier-Stokes flow simulations and monolithic fluid-structure interaction problems.

中文翻译：

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复杂物理模拟中多保真不确定性传播的广义概率学习方法

不确定性传播的两个最重要挑战与复杂物理模型仿真的高计算成本和随机输入的高维有关。在实际应用中，会遇到这两个问题，不确定性量化的标准方法要么失败，要么不可行。为了克服当前的局限性，我们提出了一种概率多保真框架，可以在小数据范围内利用原始问题的低保真模型版本。该方法通过学习高保真模型和低保真模型的输出之间的依赖关系，而不是明确考虑高维输入，从而规避了维度灾难。我们用随机输入的一组低维信息特征来补充低保真模型提供的信息，这些特征是通过结合使用有监督和无监督的降维技术来发现的。我们分析的目标是对高保真模型的完整概率响应进行有效和准确的估计。尽管低保真预测器提供了不完整和嘈杂的信息，但我们证明了可以在小数据范围内获得对感兴趣数量的准确和可证明的估计，即，运行的高保真模型显着少于当前状态- 不确定性传播的艺术方法。