# Algorithmic Linearly Constrained Gaussian Processes

arXiv: https://arxiv.org/abs/1801.09197 · HTML: https://mlsanity.com/paper/1801.09197/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2018.

## Built on by (most cited first, 20 of 20)

- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
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- [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)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [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)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Gaussian Process Regression for Inverse Problems in Linear PDEs](https://mlsanity.com/paper/2502.04276/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Hierarchical Non-Stationary Temporal Gaussian Processes With L1-Regularization](https://mlsanity.com/paper/2105.09695/) (2021)
- [Efficient methods for Gaussian Markov random fields under sparse linear constraints](https://mlsanity.com/paper/2106.01712/) (2021)
- [Physics-informed Gaussian Processes as Linear Model Predictive Controller](https://mlsanity.com/paper/2412.04502/) (2024)
- [Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity](https://mlsanity.com/paper/2511.04518/) (2025)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)
- [Multi-output Gaussian process prediction of physical fields under linear equality constraints](https://mlsanity.com/paper/2608.25709/) (2026)

## Abstract (arXiv)

We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr"obner bases. If successful, a push forward Gaussian process along the paramerization is the desired prior. We consider several examples from physics, geomathematics and control, among them the full inhomogeneous system of Maxwell's equations. By bringing together stochastic learning and computer algebra in a novel way, we combine noisy observations with precise algebraic computations.