mlsanityData to 7 Oct 2026

Physics-Informed Variational State-Space Gaussian Processes

arXiv 2409.13876 ↗First posted 20 Sep 2024cs.LG stat.MLcited by 7

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Abstract (arXiv)

Differential equations are important mechanistic models that are integral to many scientific and engineering applications. With the abundance of available data there has been a growing interest in data-driven physics-informed models. Gaussian processes (GPs) are particularly suited to this task as they can model complex, non-linear phenomena whilst incorporating prior knowledge and quantifying uncertainty. Current approaches have found some success but are limited as they either achieve poor computational scalings or focus only on the temporal setting. This work addresses these issues by introducing a variational spatio-temporal state-space GP that handles linear and non-linear physical constraints while achieving efficient linear-in-time computation costs. We demonstrate our methods in a range of synthetic and real-world settings and outperform the current state-of-the-art in both predictive and computational performance.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 40 of its 77 references that are in the map. First posted on arXiv 20 Sep 2024.

Builds on

40 of 77 references.

References · 40

  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  3. 2013Gaussian Processes for Big Datacited by 377
  4. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  5. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  6. 2017Doubly Stochastic Variational Inference for Deep Gaussian Processescited by 143
  7. 2014Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasetscited by 71
  8. 2017Conjugate-Computation Variational Inference : Converting Variational Inference in Non-Conjugate Models to Inferences in Conjugate Modelscited by 65
  9. 2024Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AIcited by 60
  10. 2018Learning unknown ODE models with Gaussian processescited by 54
  11. 2017Linearly constrained Gaussian processescited by 41
  12. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
28 more
  1. 2018Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Modelscited by 38
  2. 2021Spatio-Temporal Variational Gaussian Processescited by 32
  3. 2018Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspectivecited by 31
  4. 2019Temporal Parallelization of Bayesian Smootherscited by 31
  5. 2020Handling the Positive-Definite Constraint in the Bayesian Learning Rulecited by 31
  6. 2015Modeling and interpolation of the ambient magnetic field by Gaussian processescited by 28
  7. 2018Scaling Gaussian Process Regression with Derivativescited by 27
  8. 2018Heterogeneous Multi-output Gaussian Process Predictioncited by 26
  9. 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22
  10. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  11. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
  12. 2020Fast Variational Learning in State-Space Gaussian Process Modelscited by 20
  13. 2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20
  14. 2017Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problemscited by 19
  15. 2020Stable Implementation of Probabilistic ODE Solverscited by 19
  16. 2019Stein's Lemma for the Reparameterization Trick with Exponential Family Mixturescited by 18
  17. 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
  18. 2021Scaling Gaussian Processes with Derivative Information Using Variational Inferencecited by 15
  19. 2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
  20. 2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
  21. 2021Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guaranteescited by 12
  22. 2023Gaussian processes at the Helm(holtz): A more fluid model for ocean currentscited by 11
  23. 2020Multi-class Gaussian Process Classification with Noisy Inputscited by 10
  24. 2019Variational Bayes on Manifoldscited by 9
  25. 2021Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunitiescited by 7
  26. 2024Conformalized Physics-Informed Neural Networkscited by 7
  27. 2021Low-rank statistical finite elements for scalable model-data synthesiscited by 3
  28. 2018Additive multivariate Gaussian processes for joint species distribution modeling with heterogeneous datacited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 2; 2026: 5.

7 papers.

  1. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  2. 2025Position: Biology is the Challenge Physics-Informed ML Needs to Evolvecited by 4
  3. 2026Conditioning Gaussian Processes on Almost Anythingcited by 3
  4. 2025BALLAST: Bayesian Active Learning with Look-ahead Amendment for Sea-drifter Trajectories under Spatio-Temporal Vector Fieldscited by 1
  5. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  6. 2026Spatio-temporal probabilistic forecast using MMAF-guided learningcited by 0
  7. 2026Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processescited by 0