1,261 research outputs found
Manifold learning for the emulation of spatial fields from computational models
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
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Manifold learning for emulations of computer models
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
Emulating dynamic non-linear simulators using Gaussian processes
The dynamic emulation of non-linear deterministic computer codes where the
output is a time series, possibly multivariate, is examined. Such computer
models simulate the evolution of some real-world phenomenon over time, for
example models of the climate or the functioning of the human brain. The models
we are interested in are highly non-linear and exhibit tipping points,
bifurcations and chaotic behaviour. However, each simulation run could be too
time-consuming to perform analyses that require many runs, including
quantifying the variation in model output with respect to changes in the
inputs. Therefore, Gaussian process emulators are used to approximate the
output of the code. To do this, the flow map of the system under study is
emulated over a short time period. Then, it is used in an iterative way to
predict the whole time series. A number of ways are proposed to take into
account the uncertainty of inputs to the emulators, after fixed initial
conditions, and the correlation between them through the time series. The
methodology is illustrated with two examples: the highly non-linear dynamical
systems described by the Lorenz and Van der Pol equations. In both cases, the
predictive performance is relatively high and the measure of uncertainty
provided by the method reflects the extent of predictability in each system
Fast emulation of anisotropies induced in the cosmic microwave background by cosmic strings
Cosmic strings are linear topological defects that may have been produced
during symmetry-breaking phase transitions in the very early Universe. In an
expanding Universe the existence of causally separate regions prevents such
symmetries from being broken uniformly, with a network of cosmic string
inevitably forming as a result. To faithfully generate observables of such
processes requires computationally expensive numerical simulations, which
prohibits many types of analyses. We propose a technique to instead rapidly
emulate observables, thus circumventing simulation. Emulation is a form of
generative modelling, often built upon a machine learning backbone. End-to-end
emulation often fails due to high dimensionality and insufficient training
data. Consequently, it is common to instead emulate a latent representation
from which observables may readily be synthesised. Wavelet phase harmonics are
an excellent latent representations for cosmological fields, both as a summary
statistic and for emulation, since they do not require training and are highly
sensitive to non-Gaussian information. Leveraging wavelet phase harmonics as a
latent representation, we develop techniques to emulate string induced CMB
anisotropies over a 7.2 degree field of view, with sub-arcminute resolution, in
under a minute on a single GPU. Beyond generating high fidelity emulations, we
provide a technique to ensure these observables are distributed correctly,
providing a more representative ensemble of samples. The statistics of our
emulations are commensurate with those calculated on comprehensive Nambu-Goto
simulations. Our findings indicate these fast emulation approaches may be
suitable for wide use in, e.g., simulation based inference pipelines. We make
our code available to the community so that researchers may rapidly emulate
cosmic string induced CMB anisotropies for their own analysis
A surrogate modelling approach based on nonlinear dimension reduction for uncertainty quantification in groundwater flow models
In this paper, we develop a surrogate modelling approach for capturing the output field (e.g., the pressure head) from groundwater flow models involving a stochastic input field (e.g., the hy- draulic conductivity). We use a Karhunen-Lo`eve expansion for a log-normally distributed input field, and apply manifold learning (local tangent space alignment) to perform Gaussian process Bayesian inference using Hamiltonian Monte Carlo in an abstract feature space, yielding outputs for arbitrary unseen inputs. We also develop a framework for forward uncertainty quantification in such problems, including analytical approximations of the mean of the marginalized distri- bution (with respect to the inputs). To sample from the distribution we present Monte Carlo approach. Two examples are presented to demonstrate the accuracy of our approach: a Darcy flow model with contaminant transport in 2-d and a Richards equation model in 3-d
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