This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. The first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity.

中文翻译：

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在 Grassmann 流形上使用高斯过程回归的高维模型的数据驱动代理

本文介绍了一种基于 Grassmannian 流形学习的替代建模方案，用于高维随机系统的经济高效预测。该方法通过将每个解投影到 Grassmann 流形上来利用每个解的子空间结构特征。该方法利用解决方案聚类方法来识别参数空间的区域，在这些区域上的解决方案足够相似，以便它们可以在格拉斯曼上进行插值。在此聚类中，使用正确定义的 Grassmannian 核的特征结构将降阶解划分为 Grassmann 流形上不相交的聚类，并估计每个聚类的 Karcher 均值。然后，使用指数映射将每个簇中的点投影到原点位于相应 Karcher 均值的切线空间上。对于每个集群，训练高斯过程回归模型，将系统的输入参数映射到投影到切线空间的相应集群的简化解点。使用这种高斯过程模型，可以在参数空间中的任何新点有效地预测全场解。在某些情况下，解决方案集群将跨越参数空间的不相交区域。在这种情况下，对于每个解决方案集群，我们利用第二个基于密度的空间集群来对欧几里德空间中的相应输入参数点进行分组。所提出的方法应用于两个数值例子。第一个是初始条件不确定的非线性随机常微分方程。第二个涉及使用塑性的剪切变换区理论对模型非晶态固体中的塑性变形进行建模。