A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness
Huan Lei Pacific Northwest National Laboratory, Richland, WA 99352. Jing Li Pacific Northwest National Laboratory, Richland, WA 99352. Peiyuan Gao Pacific Northwest National Laboratory, Richland, WA 99352. Panos Stinis Pacific Northwest National Laboratory, Richland, WA 99352. Department of Applied Mathematics, University of Washington, Seattle, WA 98195. Nathan A. Baker Pacific Northwest National Laboratory, Richland, WA 99352. Division of Applied Mathematics, Brown University, Providence, RI 02912.
Abstract
The challenge of quantifying uncertainty propagation in real-world systems is rooted in the high-dimensionality of the stochastic input and the frequent lack of explicit knowledge of its probability distribution. Traditional approaches show limitations for such problems, especially when the size of the training data is limited. To address these difficulties, we have developed a general framework of constructing surrogate models on spaces of stochastic input with arbitrary probability measure irrespective of the mutual dependencies between individual components of the random inputs and the analytical form. The present Data-driven Sparsity-enhancing Rotation for Arbitrary Randomness (DSRAR) framework includes a data-driven construction of multivariate polynomial basis for arbitrary mutually dependent probability measure and a sparsity enhancement rotation procedure. This sparsity enhancement method was initially proposed in our previous work Lei_Yang_MMS_2015 for Gaussian density distributions, which may not be feasible for non-Gaussian distributions due to the loss of orthogonality after the rotation. To remedy such difficulties, we developed a new data-driven approach to construct
中文速览
真实系统中的不确定性量化（Uncertainty Quantification, UQ）面临两大难题：随机输入维度极高，且其概率分布往往未知、各分量之间存在复杂依赖关系。针对这两个痛点，作者提出了一套名为 DSRAR（数据驱动稀疏增强旋转，Data-driven Sparsity-enhancing Rotation for Arbitrary Randomness）的通用框架，核心创新在于：先用数据驱动的方式为任意相互依赖的随机输入构造多元正交多项式基（amdP），再通过旋转变换进一步增强代理模型展开系数的稀疏性，最后借助压缩感知（Compressed Sensing）在训练数据极为有限的情况下准确恢复稀疏表示。在偏微分方程（PDE）和高维构象空间（约十维）分子系统等挑战性问题上的测试表明，该方法显著优于直接套用传统多项式混沌展开的做法，对非高斯、分布未知或仅以样本集隐式给定的随机输入均有效。这项工作为现实科学与工程问题中处理复杂随机输入的不确定性传播提供了一个严格且实用的计算框架。
原文 arXiv:1804.08609;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1804.08609v4