Data-driven projection pursuit adaptation of polynomial chaos expansions for dependent high-dimensional parameters

Uncertainty quantification (UQ) and inference involving a large number of parameters are valuable tools for problems associated with heterogeneous and non-stationary behaviors. The difficulty with these problems is exacerbated when these parameters are statistically dependent requiring statistical characterization over joint measures. Probabilistic modeling methodologies stand as effective tools in the realms of UQ and inference. Among these, polynomial chaos expansions (PCE), when adapted to low-dimensional quantities of interest (QoI), provide effective yet accurate approximations for these QoI in terms of an adapted orthogonal basis. These adaptation techniques have been cast as projection pursuits in Gaussian Hilbert space in what has been referred to as a projection pursuit adaptation (PPA) by Xiaoshu Zeng and Roger Ghanem (2023). The PPA method efficiently identifies an optimal low-dimensional space for representing the QoI and simultaneously evaluates an optimal PCE within that space. The quality of this approximation clearly depends on the size of the training dataset, which is typically a function of the adapted reduced dimension. The complexity of the problem is thus mediated by the complexity of the low-dimensional quantity of interest and not the complexity of the high-dimensional parameter space. In this paper, our objective is to tackle the challenge of dependent parameters while constructing the PPA, utilizing a generative data-driven framework that requires a fixed number of pre-evaluated (parameter, QoI) pairs. While PCE approaches dealing with dependent input parameters have already been introduced by Christian Soize and Roger Ghanem (2004) their coupling with basis adaptation remains an outstanding task without which they remain plagued by the curse of dimensionality. For modest-sized parameters, mapping such as the Rosenblatt transformation can be employed to decouple the dependent variables. This strategy requires access to the joint distribution of the random variables which is usually lacking, requiring significantly more data than is typically available. To overcome these limitations, we propose leveraging multivariate Regular Vine (R-vine) copulas to encapsulate the dependency structure within parameters, manifested as a joint cumulative density function (CDF). The Rosenblatt transformation can then be applied to decouple the dependent input data, mapping them to samples from independent Gaussian variables. Conversely, we can generate dependent samples from independent Gaussian variables while maintaining the learned dependencies. This generative capability ensures that the reconstructed dependency structure is faithfully preserved in the generated samples. Endowed with the ability to diagonalize measures on product spaces, the R-vine copula blends seamlessly with the PPA method, resulting in a unified procedure for constructing optimally reduced PCE models tailored for high-dimensional problems with dependent parameter spaces. The proposed methodology attains remarkable accuracy for both UQ and inference. In the latter, the constructed PCE model adeptly serves as a generative and convergent surrogate model for machine learning regression. The efficiency of the proposed methodology is validated through two distinct applications: water flow through a borehole and structural dynamics.

不确定性定量（UQ）和涉及大量参数的推理是与异质和非平稳行为相关的问题的有价值工具。当这些参数在统计上取决于需要统计表征对关节度量的统计表征时，这些问题的困难会加剧。概率建模方法在UQ和推理领域中是有效的工具。其中，多项式混乱扩展（PCE）在适应低维的兴趣（QOI）时，以适应性的正交基础为这些QOI提供有效但准确的近似值。这些适应技术已被作为高斯希尔伯特（Hilbert Space）的投射追求，在被称为投影追求适应（PPA）的投影中，由Xiaoshu Zeng和Roger Ghanem（2023）（2023）。 PPA方法有效地识别了代表QOI的最佳低维空间，并同时评估该空间内的最佳PCE。该近似值的质量显然取决于训练数据集的大小，这通常是适应性减小的尺寸的函数。因此，问题的复杂性是由低维量的复杂性介导的，而不是高维参数空间的复杂性。 在本文中，我们的目标是在构建PPA时应对依赖参数的挑战，并利用生成数据驱动的框架，该框架需要固定数量的预评级（参数，QOI）对。虽然基督教Soize和Roger Ghanem（2004）已经引入了与依赖输入参数有关的PCE方法，但他们与基础适应性的耦合仍然是一项杰出的任务，而没有这些任务，它们仍然受到维度的诅咒的困扰。对于适中的参数，可以使用诸如Rosenblatt转换之类的映射来解除因变量。该策略需要访问通常缺少的随机变量的联合分布，需要比通常可用的数据要多得多。为了克服这些局限性，我们提出了利用多元常规葡萄藤（R-Vine）Copulas将依赖关系结构封装在参数内的依赖性结构，这表现为关节累积密度函数（CDF）。然后可以应用Rosenblatt转换将相关输入数据分解为将其映射到独立高斯变量的样品中。相反，我们可以从独立的高斯变量中生成依赖样本，同时保持学习的依赖性。这种生成能力确保了重建的依赖结构在生成的样品中忠实保存。 R-Vine Copula具有对对角度的测量能力，与PPA方法无缝融合，从而制定了统一的程序，用于构建针对依赖参数空间的高维问题定制的最佳降低PCE模型。所提出的方法对UQ和推断都具有显着的准确性。在后者中，构造的PCE模型熟练地用作机器学习回归的生成和收敛替代模型。提出的方法的效率通过两种不同的应用来验证：水流通过钻孔和结构动力学。