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 measures and a sparsity enhancement rotation procedure. This sparsity-enhancing rotation method was initially proposed in our previous work [1] 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 orthonormal polynomials for arbitrary mutually dependent randomness, ensuring the constructed basis maintains the orthogonality/near-orthogonality with respect to the density of the rotated random vector, where directly applying the regular polynomial chaos including arbitrary polynomial chaos (aPC) [2] shows limitations due to the assumption of the mutual independence between the components of the random inputs. The developed DSRAR framework leads to accurate recovery, with only limited training data, of a sparse representation of the target functions. The effectiveness of our method is demonstrated in challenging problems such as partial differential equations and realistic molecular systems within high-dimensional (O(10)) conformational spaces where the underlying density is implicitly represented by a large collection of sample data, as well as systems with explicitly given non-Gaussian probabilistic measures.

中文翻译：

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具有任意相互依赖随机性的稀疏增强代理的数据驱动框架

量化现实世界系统中不确定性传播的挑战源于随机输入的高维性以及经常缺乏对其概率分布的明确了解。传统方法对此类问题有局限性，尤其是在训练数据规模有限的情况下。为了解决这些困难，我们开发了一个通用框架，用于在具有任意概率度量的随机输入空间上构建代理模型，而不管随机输入的各个组件与分析形式之间的相互依赖关系。本数据驱动的任意随机性稀疏增强旋转（DSRAR）框架包括用于任意相互依赖概率度量的多元多项式基的数据驱动构造和稀疏增强旋转过程。这种稀疏增强旋转方法最初是在我们之前的工作 [1] 中针对高斯密度分布提出的，由于旋转后正交性丢失，这对于非高斯分布可能不可行。为了解决这些困难，我们开发了一种新的数据驱动方法来构造任意相互依赖随机性的正交多项式，确保构造的基础保持相对于旋转随机向量密度的正交性/近正交性，其中直接应用包括任意多项式混沌 (aPC) [2] 在内的规则多项式混沌显示出由于随机输入分量之间相互独立的假设而产生的局限性。开发的 DSRAR 框架导致目标函数的稀疏表示的准确恢复，只有有限的训练数据。我们方法的有效性在高维（O（10））构象空间中的偏微分方程和现实分子系统等具有挑战性的问题中得到证明，其中潜在密度由大量样本数据以及系统隐式表示使用明确给出的非高斯概率度量。