Manifold learning for the emulation of spatial fields from computational models

JOURNAL OF COMPUTATIONAL PHYSICS

卷:326

Article

Xing, W. W.¹  Triantafyllidis, V.¹  Shah, A. A.¹  Nair, P. B.²  Zabaras, N.^1,3 

[1] Univ Warwick, Warwick Ctr Predict Modelling, Coventry CV4 7AL, W Midlands, England

[2] Univ Toronto, Inst Aerosp Studies, 4925 Dufferin St, Toronto, ON M3H 5T6, Canada

[3] Univ Notre Dame, Dept Aerosp & Mech Engn, 365 Fitzpatrick Hall Engn, Notre Dame, IN 46556 USA

关键词: Parameterized partial differential equations;    Gaussian process emulation;    High dimensionality;    Manifold learning;    Inverse mapping;    Kernel PCA;    Diffusion maps;   

DOI  :  10.1016/j.jcp.2016.07.040

来源: Elsevier

【 摘 要 】

Repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible. For partial differential equation (PDE) models, the outputs of interest are often spatial fields leading to high-dimensional output spaces. Although emulators can be used to find faithful and computationally inexpensive approximations of computer models, there are few methods for handling high-dimensional output spaces. For Gaussian process (GP) emulation, approximations of the correlation structure and/or dimensionality reduction are necessary. Linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies. Manifold learning can overcome the limitations of linear methods if an accurate inverse map is available. In this paper, we use kernel PCA and diffusion maps to construct GP emulators for very high-dimensional output spaces arising from PDE model simulations. For diffusion maps we develop a new inverse map approximation. Several examples are presented to demonstrate the accuracy of our approach. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free