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Title: Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold

Abstract

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. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. 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 certainmore » 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. Here, the first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.« less

Authors:
ORCiD logo [1];  [1]
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  1. Johns Hopkins Univ., Baltimore, MD (United States)
Publication Date:
Research Org.:
Johns Hopkins Univ., Baltimore, MD (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1853643
Alternate Identifier(s):
OSTI ID: 1638520
Grant/Contract Number:  
SC0020428
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 370; Journal Issue: C; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; Engineering; Mathematics; Mechanics; Grassmann manifold; Spectral clustering; Gaussian process regression; Machine learning; Nonlinear projection; Interpolation

Citation Formats

Giovanis, D. G., and Shields, M. D. Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold. United States: N. p., 2020. Web. doi:10.1016/j.cma.2020.113269.
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Giovanis, D. G., & Shields, M. D. Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold. United States. https://doi.org/10.1016/j.cma.2020.113269
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Giovanis, D. G., and Shields, M. D. Wed . "Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold". United States. https://doi.org/10.1016/j.cma.2020.113269. https://www.osti.gov/servlets/purl/1853643.
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@article{osti_1853643,
title = {Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold},
author = {Giovanis, D. G. and Shields, M. D.},
abstractNote = {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. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. 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. Here, the first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.},
doi = {10.1016/j.cma.2020.113269},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 370,
place = {United States},
year = {Wed Jul 15 00:00:00 EDT 2020},
month = {Wed Jul 15 00:00:00 EDT 2020}
}
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    Works referencing / citing this record:

    Grassmannian diffusion maps based dimension reduction and classification for high-dimensional data
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    Simpler Grassmannian optimization
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