In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemanniann K-means and the minimization of the sample Frechet variance on the Grassmann manifold to identify "local" principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Benard convection problem

翻译：本文提出了一种基于流形学习的代理建模框架，用于高维随机系统的不确定性量化。首要目标是对现有模拟数据进行数据挖掘，识别一组低维（潜在）描述符，从而高效参数化高维计算模型的响应。为此，我们采用响应格拉斯曼流形上的主测地线分析，识别出一组可能具有不同维度的互斥主测地线子流形，以捕获数据中的变异性。由于格拉斯曼流形上的操作要求数据集中，我们提出一种基于黎曼K均值聚类与格拉斯曼流形上样本弗雷歇方差最小化的自适应算法，用于识别代表参数空间不同系统行为的“局部”主测地线子流形。随后，利用多项式混沌展开构建随机输入参数与响应在这些局部主测地线子流形上投影之间的映射。该方法在四个测试案例中得到验证：包含超球面上点的简单示例、Lotka-Volterra动力系统、连续搅拌釜式化学反应器系统以及二维瑞利-贝纳德对流问题。