Advanced Stationary and Non-Stationary Kernel Designs for Domain-Aware Gaussian Processes

TL;DR

This paper introduces advanced kernel designs for Gaussian processes that incorporate domain-specific characteristics like symmetry, periodicity, and non-stationarity, enhancing function approximation accuracy in scientific applications.

Contribution

It proposes novel kernel constructions that embed physical domain knowledge into Gaussian process models, improving their flexibility and relevance.

Findings

Enhanced accuracy in function approximation with domain-aware kernels

Significant improvement over traditional kernels in scientific datasets

Demonstrated benefits of incorporating physical constraints into Gaussian processes

Abstract

Gaussian process regression is a widely-applied method for function approximation and uncertainty quantification. The technique has gained popularity recently 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, non-stationary kernel designs can be defined in the same framework to yield flexible multi-task 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 including domain knowledge, communicated through advanced kernel designs, has a significant impact on the accuracy and relevance of the function approximation.