Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification

TL;DR

A manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems and proposes 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.

Abstract

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

Figures (16)

• Figure 1: For a set of training images that correspond to three different classes, namely cats, dogs, and spiders, we can classify these images based on their respective subspaces on the Grassmannian and when a new image (e.g., an image of a dog) is introduced we can assign it to the appropriate class.
• Figure 2: Illustration of the proposed two-phase dimension reduction scheme using Grassmann manifold projection and principal geodesic analysis. The strain field images of computational head models are adopted from upadhyay2022data. Each circle on the manifold corresponds to a realization of the strain field. Distinct colors represent different clusters on the Grassmann.
• Figure 3: Points on the hypersphere: (a) Realizations of $\boldsymbol{\theta}$. (b) Corresponding realizations of the responses $\mathbf{y}$. (c) Representation of the points on the Stiefel manifold $\mathbb{S}(1, 3)$
• Figure 4: Convergence of the Silhouette coefficient for an increasing number of clusters and for training data sets of increasing size $\mathcal{N}$. The optimum number of clusters is the one corresponding to the maximum peak of the each plot.
• Figure 5: Points on the hypersphere: (a) Cluster of points on $\mathbb{S}(1, 3)$; for two clusters we plot the geodesics on the sphere corresponding to the first and second principal components (b) 150 PCE predicted points on $\mathbb{S}(1, 3)$, (c) and the corresponding solutions in Cartesian space.