mlsanityData to 7 Oct 2026

Bayesian Probabilistic Numerical Methods

arXiv 1702.03673 ↗First posted 13 Feb 2017stat.ME cs.NA math.NAcited by 50

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Abstract (arXiv)

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with a challenging industrial application presented.

Key facts. Cited by 50 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Feb 2017.

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Citing papers per year: 2021: 15; 2022: 6; 2023: 6; 2024: 10; 2025: 7; 2026: 6.

50 papers.

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  1. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  2. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  3. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  4. 2021Computational Graph Completioncited by 28
  5. 2022Posterior and Computational Uncertainty in Gaussian Processescited by 23
  6. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  7. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  8. 2022Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posedcited by 16
  9. 2021Bayesian Numerical Methods for Nonlinear Partial Differential Equationscited by 15
  10. 2021Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equationscited by 13
  11. 2023A spectrum of physics-informed Gaussian processes for regression in engineeringcited by 13
  12. 2024Computation-Aware Kalman Filtering and Smoothingcited by 12
38 more
  1. 2022Fenrir: Physics-Enhanced Regression for Initial Value Problemscited by 11
  2. 2024FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learningcited by 11
  3. 2021ProbNum: Probabilistic Numerics in Pythoncited by 10
  4. 2021The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learningcited by 9
  5. 2022Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spacescited by 9
  6. 2024Conditional Bayesian Quadraturecited by 8
  7. 2024Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inferencecited by 8
  8. 2021Pick-and-Mix Information Operators for Probabilistic ODE Solverscited by 6
  9. 2024Calibrated Computation-Aware Gaussian Processescited by 6
  10. 2025Conformal Prediction as Bayesian Quadraturecited by 6
  11. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  12. 2021Bayesian Quadrature on Riemannian Data Manifoldscited by 5
  13. 2021Small Sample Spaces for Gaussian Processescited by 5
  14. 2023Probabilistic Exponential Integratorscited by 4
  15. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  16. 2021Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMCcited by 3
  17. 2023Bayesian Numerical Integration with Neural Networkscited by 3
  18. 2021Invariant Priors for Bayesian Quadraturecited by 2
  19. 2023Accelerating Non-Conjugate Gaussian Processes By Trading Off Computation For Uncertaintycited by 2
  20. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  21. 2021Tractable Computation of Expected Kernelscited by 1
  22. 2021Linear-Time Probabilistic Solutions of Boundary Value Problemscited by 1
  23. 2024Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equationscited by 1
  24. 2024Impact of Computation in Integral Reinforcement Learning for Continuous-Time Controlcited by 1
  25. 2025Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fieldscited by 1
  26. 2025Online Conformal Probabilistic Numerics via Adaptive Edge-Cloud Offloadingcited by 1
  27. 2025Physics-based deep kernel learning for parameter estimation in high dimensional PDEscited by 1
  28. 2026Bayesian Quadraturecited by 1
  29. 2021A probabilistic Taylor expansion with Gaussian processescited by 0
  30. 2021Black Box Probabilistic Numericscited by 0
  31. 2024Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirementscited by 0
  32. 2025Constructive Disintegration and Conditional Modescited by 0
  33. 2025BayesSum: Bayesian Quadrature in Discrete Spacescited by 0
  34. 2026FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivativescited by 0
  35. 2026Ensemble-Conditional Gaussian Processes (Ens-CGP): Representation, Geometry, and Inferencecited by 0
  36. 2026Multifidelity Gaussian process regression for solving nonlinear partial differential equationscited by 0
  37. 2026Scalable Bayesian Inference for Nonlinear Conservation Lawscited by 0
  38. 2026Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEscited by 0