Manifold learning for emulations of computer models

Abstract

Computer simulations are widely used in scientific research and engineering areas. Thought they could provide accurate result, the computational expense is normally high and thus hinder their applications to problems, where repeated evaluations are required, e.g, design optimization and uncertainty quantification. 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 thesis, manifold learning is applied to construct GP emulators for very high-dimensional output spaces arising from parameterised PDE model simulations. Artificial neural network (ANN) support vector machine (SVM) emulators using manifold learning are also studied. A general framework for the inverse map approximation and a new efficient method for diffusion maps were developed. The manifold learning based emulators are then to extend reduced order models (ROMs) based on proper orthogonal decomposition to dynamic, parameterized PDEs. A similar approach is used to extend the discrete empirical interpolation method (DEIM) to ROMs for nonlinear, parameterized dynamic PDEs