The present work addresses the issue of accurate stochastic approximations in high-dimensional parametric space using tools from uncertainty quantification (UQ). The basis adaptation method and its accelerated algorithm in polynomial chaos expansions (PCE) were recently proposed to construct low-dimensional approximations adapted to specific quantities of interest (QoI). The present paper addresses one difficulty with these adaptations, namely their reliance on quadrature point sampling, which limits the reusability of potentially expensive samples. Projection pursuit (PP) is a statistical tool to find the ``interesting'' projections in high-dimensional data and thus bypass the curse-of-dimensionality. In the present work, we combine the fundamental ideas of basis adaptation and projection pursuit regression (PPR) to propose a novel method to simultaneously learn the optimal low-dimensional spaces and PCE representation from given data. While this projection pursuit adaptation (PPA) can be entirely data-driven, the constructed approximation exhibits mean-square convergence to the solution of an underlying governing equation and is thus subject to the same physics constraints. The proposed approach is demonstrated on a borehole problem and a structural dynamics problem, demonstrating the versatility of the method and its ability to discover low-dimensional manifolds with high accuracy with limited data. In addition, the method can learn surrogate models for different quantities of interest while reusing the same data set.
翻译:目前的工作涉及利用不确定性量化工具(UQ)在高维参数空间进行准确的近似孔径测量的问题。最近提议采用基础适应方法及其在多元混乱扩张中加速算法,以构建适合具体兴趣数量的低维近似值(QoI)。本文件涉及这些调整的一个困难,即它们依赖四点取样,这限制了潜在昂贵样品的可重复性。预测跟踪(PP)是一个统计工具,用来查找“有兴趣”在高维数据的预测,从而绕过维度的诅咒。在目前的工作中,我们将基础适应和预测追求回归的基本想法(PPR)结合起来,以提出一种新颖的方法,同时学习最佳低维度空间和从特定数据得到的PCE代表。虽然这种预测跟踪适应(PPA)可以完全以数据为动力,但构建的近似物展览与基本治理方程式的解决办法相趋同,因此受到同样的物理限制。拟议的方法在钻孔问题和结构动力回退缩中展示了基础问题和结构回溯回归能力,同时展示其多维性数据再研究方法的特性。