# Linearly constrained Gaussian processes

arXiv: https://arxiv.org/abs/1703.00787 · HTML: https://mlsanity.com/paper/1703.00787/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2017.

## Built on by (most cited first, 30 of 42)

- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning](https://mlsanity.com/paper/2205.06342/) (2022)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [High-Dimensional Gaussian Process Inference with Derivatives](https://mlsanity.com/paper/2102.07542/) (2021)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields](https://mlsanity.com/paper/2208.12104/) (2022)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (2022)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Learning thermodynamically constrained equations of state with uncertainty](https://mlsanity.com/paper/2306.17004/) (2023)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2406.11809/) (2024)
- [Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features](https://mlsanity.com/paper/2510.19399/) (2025)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Learning-enhanced robust controller synthesis with rigorous statistical and control-theoretic guarantees](https://mlsanity.com/paper/2105.03397/) (2021)
- [Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors](https://mlsanity.com/paper/2109.03457/) (2021)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [Adjoint-aided inference of Gaussian process driven differential equations](https://mlsanity.com/paper/2202.04589/) (2022)
- [Monotonic Gaussian process for physics-constrained machine learning with materials science applications](https://mlsanity.com/paper/2209.00628/) (2022)
- [A connection between probability, physics and neural networks](https://mlsanity.com/paper/2209.12737/) (2022)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations](https://mlsanity.com/paper/2601.21234/) (2026)
- [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)
- [Distributional Gradient Matching for Learning Uncertain Neural Dynamics Models](https://mlsanity.com/paper/2106.11609/) (2021)

## Abstract (arXiv)

We consider a modification of the covariance function in Gaussian processes to correctly account for known linear constraints. By modelling the target function as a transformation of an underlying function, the constraints are explicitly incorporated in the model such that they are guaranteed to be fulfilled by any sample drawn or prediction made. We also propose a constructive procedure for designing the transformation operator and illustrate the result on both simulated and real-data examples.