# Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features

arXiv: https://arxiv.org/abs/1904.05207 · HTML: https://mlsanity.com/paper/1904.05207/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2019.

## Built on by (most cited first, 10 of 10)

- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields](https://mlsanity.com/paper/2507.17582/) (2025)
- [BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problems](https://mlsanity.com/paper/2510.03576/) (2025)
- [Boundary Variance Inflation Causes Acquisition Bias in Gaussian Processes](https://mlsanity.com/paper/2606.07561/) (2026)
- [Spectrum Gaussian Processes Based On Tunable Basis Functions](https://mlsanity.com/paper/2107.06473/) (2021)
- [Determination of Particle-Size Distributions from Light-Scattering Measurement Using Constrained Gaussian Process Regression](https://mlsanity.com/paper/2507.03736/) (2025)

## Abstract (arXiv)

Gaussian processes (GPs) provide a powerful framework for extrapolation, interpolation, and noise removal in regression and classification. This paper considers constraining GPs to arbitrarily-shaped domains with boundary conditions. We solve a Fourier-like generalised harmonic feature representation of the GP prior in the domain of interest, which both constrains the GP and attains a low-rank representation that is used for speeding up inference. The method scales as O(nm2) in prediction and O(m3) in hyperparameter learning for regression, where n is the number of data points and m the number of features. Furthermore, we make use of the variational approach to allow the method to deal with non-Gaussian likelihoods. The experiments cover both simulated and empirical data in which the boundary conditions allow for inclusion of additional physical information.