# Linearly Constrained Gaussian Processes with Boundary Conditions

arXiv: https://arxiv.org/abs/2002.00818 · HTML: https://mlsanity.com/paper/2002.00818/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2020.

## Built on by (most cited first, 14 of 14)

- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields](https://mlsanity.com/paper/2208.12104/) (2022)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Designing Robust Biotechnological Processes Regarding Variabilities using Multi-Objective Optimization Applied to a Biopharmaceutical Seed Train Design](https://mlsanity.com/paper/2205.03261/) (2022)
- [Monotonic Gaussian process for physics-constrained machine learning with materials science applications](https://mlsanity.com/paper/2209.00628/) (2022)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Efficient methods for Gaussian Markov random fields under sparse linear constraints](https://mlsanity.com/paper/2106.01712/) (2021)
- [A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information](https://mlsanity.com/paper/2201.11023/) (2022)
- [BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problems](https://mlsanity.com/paper/2510.03576/) (2025)
- [Gaussian Process Regression with Soft Inequality and Monotonicity Constraints](https://mlsanity.com/paper/2404.02873/) (2024)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)
- [Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints](https://mlsanity.com/paper/2610.13804/) (2026)

## Abstract (arXiv)

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Gr"obner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.