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

arXiv: https://arxiv.org/abs/2102.03432 · HTML: https://mlsanity.com/paper/2102.03432/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 5 Feb 2021.

## Builds on (most cited first, 5 of 5)

- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [A Framework for Interdomain and Multioutput Gaussian Processes](https://mlsanity.com/paper/2003.01115/) (2020)
- [Autonomous Materials Discovery Driven by Gaussian Process Regression with Inhomogeneous Measurement Noise and Anisotropic Kernels](https://mlsanity.com/paper/2006.02489/) (2020)
- [Regression-based covariance functions for nonstationary spatial modeling](https://mlsanity.com/paper/1410.1494/) (2014)
- [Constrained Gaussian Process Learning for Model Predictive Control](https://mlsanity.com/paper/1911.10809/) (2019)

## Built on by (most cited first, 7 of 7)

- [Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey](https://mlsanity.com/paper/2106.08443/) (2021)
- [A Unifying Perspective on Non-Stationary Kernels for Deeper Gaussian Processes](https://mlsanity.com/paper/2309.10068/) (2023)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Towards a Science Exocortex](https://mlsanity.com/paper/2406.17809/) (2024)
- [gp2Scale: A Class of Compactly Supported Non-Stationary Kernels and Distributed Computing for Exact Gaussian Processes on 10 Million Data Points](https://mlsanity.com/paper/2512.06143/) (2025)
- [Expressive Mortality Models through Gaussian Process Kernels](https://mlsanity.com/paper/2305.01728/) (2023)
- [Physically-informed change-point kernels for structural dynamics](https://mlsanity.com/paper/2506.11625/) (2025)

## Abstract (arXiv)

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.