A novel data-driven sparse polynomial chaos expansion for high-dimensional problems based on active subspace and sparse Bayesian learning

Polynomial chaos expansion (PCE) has recently drawn growing attention in the community of stochastic uncertainty quantification (UQ). However, the drawback of the curse of dimensionality limits its application to complex and large-scale structures, and the PCE construction needs the complete knowledge of probability distributions of input variables, which may be impractical for real-world problems. To overcome these difficulties, this study proposes an active learning active subspace-based data-driven sparse PCE method (AL-AS-DDSPCE). First, we use the active subspace (AS) theory to reduce the dimension of the original input space, and establish the measure-consistent data-driven polynomial chaos bases in the reduced input space based on the samples of the original input random variables. Subsequently, to bypass the gradient calculation in the traditional AS method, we combine the sparse Bayesian learning with the manifold learning theory and propose an active learning AS method to obtain the subspace mapping matrix. The proposed AL-AS-DDSPCE can find the low-dimensional subspace of the original input space with the elaborate active learning algorithm, which does not require the probability distribution of input variables but is driven by the sample data of input random variables and the response data of the design samples, and can construct the PCE model efficiently and accurately. We verify the proposed method using two classical high-dimensional numerical examples, a 200-bar truss without explicit expression and one practical engineering problem. The results show that the AL-AS-DDSPCE is a good choice for solving high-dimensional UQ problems.