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Abstract
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Gaussian process regression is a widely applied method for function approximation
and uncertainty quantification. The technique has recently gained popularity in the
machine learning community due to its robustness and interpretability. The
mathematical methods we discuss in this paper are an extension of the Gaussian
process framework. We are proposing advanced kernel designs that only allow for
functions with certain desirable characteristics to be elements of the reproducing
kernel Hilbert space (RKHS) that underlies all kernel methods and serves as the
sample space for Gaussian process regression. These desirable characteristics reflect
the underlying physics; two obvious examples are symmetry and periodicity
constraints. In addition, we want to draw attention to nonstationary kernel
designs that can be defined in the same framework to yield flexible multitask
Gaussian processes. We will show the impact of advanced kernel designs
on Gaussian processes using several synthetic and two scientific data sets.
The results show that informing a Gaussian process of domain knowledge,
combined with additional flexibility and communicated through advanced kernel
designs, has a significant impact on the accuracy and relevance of the function
approximation.
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Keywords
Gaussian processes, machine learning, uncertainty
quantification
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Mathematical Subject Classification
Primary: 60-08, 60G15
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Milestones
Received: 12 November 2021
Revised: 8 March 2022
Accepted: 15 April 2022
Published: 7 October 2022
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