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

arXiv 2006.09319 ↗First posted 16 Jun 2020cs.LG math.ST stat.MLcited by 51

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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.

Key facts. Cited by 51 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Jun 2020.

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Citing papers per year: 2021: 5; 2022: 8; 2023: 13; 2024: 10; 2025: 8; 2026: 7.

51 papers.

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  1. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  2. 2024A review on data-driven constitutive laws for solidscited by 35
  3. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  4. 2024Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantificationcited by 17
  5. 2023Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanicscited by 16
  6. 2022Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effectscited by 15
  7. 2023Physics-Informed Polynomial Chaos Expansionscited by 15
  8. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
  9. 2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 13
  10. 2024Universal Functional Regression with Neural Operator Flowscited by 9
  11. 2021Advanced Stationary and Non-Stationary Kernel Designs for Domain-Aware Gaussian Processescited by 7
  12. 2022Constraining Gaussian processes for physics-informed acoustic emission mappingcited by 7
38 more
  1. 2023Learning thermodynamically constrained equations of state with uncertaintycited by 5
  2. 2023Enhanced Human-Robot Collaboration using Constrained Probabilistic Human-Motion Predictioncited by 5
  3. 2022Physically Meaningful Uncertainty Quantification in Probabilistic Wind Turbine Power Curve Models as a Damage Sensitive Featurecited by 4
  4. 2022Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equationscited by 4
  5. 2024Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equationscited by 4
  6. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  7. 2021Structure-Preserving Learning Using Gaussian Processes and Variational Integratorscited by 3
  8. 2023SEAL: Simultaneous Exploration and Localization in Multi-Robot Systemscited by 3
  9. 2024Measurements with Noise: Bayesian Optimization for Co-optimizing Noise and Property Discovery in Automated Experimentscited by 3
  10. 2024SymbolFit: Automatic Parametric Modeling with Symbolic Regressioncited by 3
  11. 2022Incorporating Sum Constraints into Multitask Gaussian Processescited by 2
  12. 2022Monotonic Gaussian process for physics-constrained machine learning with materials science applicationscited by 2
  13. 2022A connection between probability, physics and neural networkscited by 2
  14. 2021Hierarchical Non-Stationary Temporal Gaussian Processes With L1-Regularizationcited by 1
  15. 2022A Kernel-Based Approach for Modelling Gaussian Processes with Functional Informationcited by 1
  16. 2023Stochastic Cell Transmission Models of Traffic Networkscited by 1
  17. 2023Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysiscited by 1
  18. 2023Gaussian Processes with State-Dependent Noise for Stochastic Controlcited by 1
  19. 2025Flow-Induced Diagonal Gaussian Processescited by 1
  20. 2025Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamicscited by 1
  21. 2025Learning solutions of parameterized stiff ODEs using Gaussian processescited by 1
  22. 2026Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curvescited by 1
  23. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  24. 2021Practical, Provably-Correct Interactive Learning in the Realizable Setting: The Power of True Believerscited by 0
  25. 2023A machine learning approach to the prediction of heat-transfer coefficients in micro-channelscited by 0
  26. 2023Index-aware learning of circuitscited by 0
  27. 2023Deep Bayesian Reinforcement Learning for Spacecraft Proximity Maneuvers and Dockingcited by 0
  28. 2024PINN-BO: A Black-box Optimization Algorithm using Physics-Informed Neural Networkscited by 0
  29. 2024Gaussian Process Regression with Soft Inequality and Monotonicity Constraintscited by 0
  30. 2024SPINEX-TimeSeries: Similarity-based Predictions with Explainable Neighbors Exploration for Time Series and Forecasting Problemscited by 0
  31. 2025Determination of Particle-Size Distributions from Light-Scattering Measurement Using Constrained Gaussian Process Regressioncited by 0
  32. 2025Fast Gaussian Processes under Monotonicity Constraintscited by 0
  33. 2025Enhancing Corrosion Resistance of Aluminum Alloys Through AI and ML Modelingcited by 0
  34. 2025A Minimalist Bayesian Framework for Stochastic Optimizationcited by 0
  35. 2026Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Modelscited by 0
  36. 2026Infinite-dimensional spherical-radial decomposition for probabilistic functions, with application to constrained optimal control and Gaussian process regressioncited by 0
  37. 2026MINTS: Minimalist Thompson Samplingcited by 0
  38. 2026Multi-output Gaussian process prediction of physical fields under linear equality constraintscited by 0

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