# A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges

arXiv: https://arxiv.org/abs/2006.09319 · HTML: https://mlsanity.com/paper/2006.09319/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

## Built on by (most cited first, 30 of 52)

- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
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- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Advanced Stationary and Non-Stationary Kernel Designs for Domain-Aware Gaussian Processes](https://mlsanity.com/paper/2102.03432/) (2021)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [Enhanced Human-Robot Collaboration using Constrained Probabilistic Human-Motion Prediction](https://mlsanity.com/paper/2310.03314/) (2023)
- [Learning thermodynamically constrained equations of state with uncertainty](https://mlsanity.com/paper/2306.17004/) (2023)
- [Physically Meaningful Uncertainty Quantification in Probabilistic Wind Turbine Power Curve Models as a Damage Sensitive Feature](https://mlsanity.com/paper/2209.15579/) (2022)
- [Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2212.11880/) (2022)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation](https://mlsanity.com/paper/2405.13149/) (2024)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [SEAL: Simultaneous Exploration and Localization in Multi-Robot Systems](https://mlsanity.com/paper/2306.12623/) (2023)
- [Measurements with Noise: Bayesian Optimization for Co-optimizing Noise and Property Discovery in Automated Experiments](https://mlsanity.com/paper/2410.02717/) (2024)
- [SymbolFit: Automatic Parametric Modeling with Symbolic Regression](https://mlsanity.com/paper/2411.09851/) (2024)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (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)
- [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)
- [A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information](https://mlsanity.com/paper/2201.11023/) (2022)
- [Stochastic Cell Transmission Models of Traffic Networks](https://mlsanity.com/paper/2304.11654/) (2023)
- [Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis](https://mlsanity.com/paper/2305.13879/) (2023)

## Abstract (arXiv)

Gaussian process regression is a popular Bayesian framework for surrogate modeling of expensive data sources. As part of a broader effort in scientific machine learning, many recent works have incorporated physical constraints or other a priori information within Gaussian process regression to supplement limited data and regularize the behavior of the model. We provide an overview and survey of several classes of Gaussian process constraints, including positivity or bound constraints, monotonicity and convexity constraints, differential equation constraints provided by linear PDEs, and boundary condition constraints. We compare the strategies behind each approach as well as the differences in implementation, concluding with a discussion of the computational challenges introduced by constraints.