Polynomial dimensional decomposition (PDD) is a surrogate method originated from the ANOVA (analysis of variance) decomposition, and has shown powerful performance in uncertainty quantification (UQ) accuracy and convergence recently. However, complex high-dimensional problems result in a large number of polynomial basis functions, leading to heavy computational burden, and the probability distributions of the input random variables are indispensable for PDD modeling and UQ, which may be unavailable in practical engineering. This study establishes an adaptive data-driven subspace PDD (ADDSPDD) for high-dimensional UQ, which employs two types of data for modeling the PDD basis function and the low-dimensional subspace directly, namely, the data of input random variables and the input-response samples. Firstly, we propose a data-driven zero-entropy criterion-based maximum entropy method for reconstructing the probability density functions (PDF) of input variables. Then, with the aid of the established PDFs, a data-driven subspace PDD (DDSPDD) is proposed based on the whitening transformation. To recover the subspace of the function of interest accurately and efficiently, we put forward an approximate active subspace method (AAS) based on the Taylor expansion under some mild premises. Finally, we integrate an adaptive learning algorithm into the DDSPDD framework based on the sparse Bayesian learning theory, obtaining our ADDSPDD; thus, the real subspace and the significant PDD basis functions can be identified with limited computational budget. We validate the proposed method by using four examples, and systematically compare four existing dimension-reduction methods with the AAS. Results show that the proposed framework is effective and the AAS is a good choice when the corresponding assumptions are satisfied.

中文翻译：

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基于最大熵方法和稀疏贝叶斯学习的自适应数据驱动子空间多项式维分解，用于高维不确定性量化

多项式维数分解（PDD）是一种起源于方差分析（ANOVA）分解的替代方法，近年来在不确定性量化（UQ）精度和收敛性方面表现出了强大的性能。然而，复杂的高维问题会导致大量的多项式基函数，导致计算负担沉重，而输入随机变量的概率分布对于PDD建模和UQ来说是必不可少的，而这在实际工程中可能无法实现。本研究建立了一种针对高维UQ的自适应数据驱动子空间PDD（ADDSPDD），它采用两类数据直接对PDD基函数和低维子空间进行建模，即输入随机变量的数据和输入的数据- 响应样本。首先，我们提出了一种基于数据驱动的零熵准则的最大熵方法，用于重建输入变量的概率密度函数（PDF）。然后，借助已建立的PDF，提出了基于白化变换的数据驱动子空间PDD（DDSPDD）。为了准确有效地恢复感兴趣函数的子空间，我们在一些温和的前提下提出了一种基于泰勒展开的近似主动子空间方法（AAS）。最后，我们基于稀疏贝叶斯学习理论将自适应学习算法集成到DDSPDD框架中，得到我们的ADDSPDD；因此，可以用有限的计算预算来识别真实子空间和重要的PDD基函数。我们通过四个例子验证了所提出的方法，并系统地将四种现有的降维方法与 AAS 进行比较。结果表明，所提出的框架是有效的，并且在满足相应假设的情况下，AAS 是一个不错的选择。