In this work we introduce a manifold learning-based method for uncertainty quantification (UQ) in systems describing complex spatiotemporal processes. Our first objective is to identify the embedding of a set of high-dimensional data representing quantities of interest of the computational or analytical model. For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear dimension reduction technique which allows us to reduce the dimensionality of the data and identify meaningful geometric descriptions in a parsimonious and inexpensive manner. Polynomial chaos expansion is then used to construct a mapping between the stochastic input parameters and the diffusion coordinates of the reduced space. An adaptive clustering technique is proposed to identify an optimal number of clusters of points in the latent space. The similarity of points allows us to construct a number of geometric harmonic emulators which are finally utilized as a set of inexpensive pretrained models to perform an inverse map of realizations of latent features to the ambient space and thus perform accurate out-of-sample predictions. Thus, the proposed method acts as an encoder-decoder system which is able to automatically handle very high-dimensional data while simultaneously operating successfully in the small-data regime. The method is demonstrated on two benchmark problems and on a system of advection-diffusion-reaction equations which model a first-order chemical reaction between two species. In all test cases, the proposed method is able to achieve highly accurate approximations which ultimately lead to the significant acceleration of UQ tasks.

中文翻译：

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高维代理模型的基于流形学习的多项式混沌扩展

在这项工作中，我们介绍了一种基于流形学习的方法，用于描述复杂时空过程的系统中的不确定性量化 (UQ)。我们的第一个目标是识别一组表示计算或分析模型感兴趣的量的高维数据的嵌入。为此，我们采用了 Grassmannian 扩散图，这是一种两步非线性降维技术，它使我们能够降低数据的维数并以简约且廉价的方式识别有意义的几何描述。然后使用多项式混沌展开来构建随机输入参数和缩减空间的扩散坐标之间的映射。提出了一种自适应聚类技术来识别潜在空间中的最佳点聚类数。点的相似性使我们能够构建许多几何谐波仿真器，最终将它们用作一组廉价的预训练模型，以执行潜在特征与环境空间实现的逆映射，从而执行准确的样本外预测。因此，所提出的方法充当编码器-解码器系统，能够自动处理非常高维的数据，同时在小数据状态下成功运行。该方法在两个基准问题和一个对流-扩散-反应方程系统上进行了演示，该系统模拟了两个物种之间的一级化学反应。在所有测试用例中，所提出的方法能够实现高度准确的近似，最终导致 UQ 任务的显着加速。