We propose a non-intrusive reduced-oder modeling method based on proper orthogonal decomposition (POD) and polynomial chaos expansion (PCE) for stochastic representations in uncertainty quantification (UQ) analysis. Firstly, POD provides an optimally ordered basis from a set of selected full-order snapshots. Truncating this optimal basis, we construct a reduced-order model with undetermined coefficients. Then, PCE is utilized to approximate the coefficients of the truncated basis. In the proposed method, we construct a PCE using a non-intrusive regression-based method. Combined with the model reduction ability of POD, the proposed method efficiently provides stochastic representations in UQ analysis. To investigate the performance of the proposed method, we provide three numerical examples, i.e., a highly nonlinear analytical function with three uncertain parameters, two-dimensional (2D) heat-driven cavity flow with a stochastic boundary temperature, and 2D heat diffusion with stochastic conductivity. The results demonstrate that the proposed method significantly reduces the computational costs and storage requirements that arise due to high-dimensional physical and random spaces, while demonstrating a similar accuracy with that of the classical sparse PCE in predicting statistical quantities. Furthermore, the proposed method reasonably predicts the outputs of the full order model using only a few snapshots.

我们提出了一种基于固有正交分解（POD）和多项式混沌展开（PCE）的非侵入式减法建模方法，用于不确定性量化（UQ）分析中的随机表示。首先，POD从一组选定的全序快照中提供最佳排序的基础。截断这个最佳基础，我们构造了系数不确定的降阶模型。然后，利用PCE近似截断后的系数。在提出的方法中，我们使用基于非侵入式回归的方法构造了PCE。结合POD的模型简化能力，该方法有效地提供了UQ分析中的随机表示。为了研究所提出方法的性能，我们提供了三个数值示例，即 具有三个不确定参数的高度非线性分析函数：具有随机边界温度的二维（2D）热驱动腔流和具有随机电导率的2D热扩散。结果表明，该方法大大降低了由于高维物理空间和随机空间而引起的计算成本和存储需求，同时在预测统计量方面显示了与经典稀疏PCE相似的准确性。此外，所提出的方法仅使用几个快照就可以合理地预测全订单模型的输出。结果表明，该方法大大降低了由于高维物理空间和随机空间而引起的计算成本和存储需求，同时在预测统计量方面显示了与经典稀疏PCE相似的准确性。此外，所提出的方法仅使用几个快照就可以合理地预测全订单模型的输出。结果表明，该方法大大降低了由于高维物理空间和随机空间而引起的计算成本和存储需求，同时在预测统计量方面显示了与经典稀疏PCE相似的准确性。此外，所提出的方法仅使用几个快照就可以合理地预测全订单模型的输出。