mlsanityData to 7 Oct 2026

Probabilistic Numerics and Uncertainty in Computations

arXiv 1506.01326 ↗First posted 3 Jun 2015math.NA cs.AI cs.LGcited by 59

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Abstract (arXiv)

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

Key facts. Cited by 59 papers in the map (data through 7 Oct 2026). First posted on arXiv 3 Jun 2015.

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Citing papers per year: 2021: 18; 2022: 8; 2023: 10; 2024: 8; 2025: 6; 2026: 9.

59 papers.

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  1. 2022Message Passing Neural PDE Solverscited by 368
  2. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 140
  3. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  4. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  5. 2021Computational Graph Completioncited by 28
  6. 2021Emulation of physical processes with Emukitcited by 24
  7. 2022Posterior and Computational Uncertainty in Gaussian Processescited by 23
  8. 2021A Probabilistic Interpretation of Self-Paced Learning with Applications to Reinforcement Learningcited by 21
  9. 2022Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombinationcited by 20
  10. 2021High-Dimensional Gaussian Process Inference with Derivativescited by 16
  11. 2021Bayesian Numerical Methods for Nonlinear Partial Differential Equationscited by 15
  12. 2021Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guaranteescited by 12
38 more
  1. 2024Computation-Aware Kalman Filtering and Smoothingcited by 12
  2. 2022Fenrir: Physics-Enhanced Regression for Initial Value Problemscited by 11
  3. 2021ProbNum: Probabilistic Numerics in Pythoncited by 10
  4. 2022Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlationscited by 10
  5. 2024Conditional Bayesian Quadraturecited by 8
  6. 2024Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inferencecited by 8
  7. 2021Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunitiescited by 7
  8. 2021Pick-and-Mix Information Operators for Probabilistic ODE Solverscited by 6
  9. 2023Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approachcited by 6
  10. 2021Bayesian Quadrature on Riemannian Data Manifoldscited by 5
  11. 2026BayeSQP: Bayesian Optimization through Sequential Quadratic Programmingcited by 5
  12. 2023Probabilistic Exponential Integratorscited by 4
  13. 2023Implicit Manifold Gaussian Process Regressioncited by 4
  14. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  15. 2021Advancing Trajectory Optimization with Approximate Inference: Exploration, Covariance Control and Adaptive Riskcited by 3
  16. 2021Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMCcited by 3
  17. 2021Efficient Stochastic Optimal Control through Approximate Bayesian Input Inferencecited by 3
  18. 2022PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equationscited by 3
  19. 2023Bayesian Numerical Integration with Neural Networkscited by 3
  20. 2023Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equationscited by 3
  21. 2023Parallel-in-Time Probabilistic Numerical ODE Solverscited by 3
  22. 2024A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Liftingcited by 3
  23. 2024Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanicscited by 3
  24. 2021Invariant Priors for Bayesian Quadraturecited by 2
  25. 2022Adaptive Cholesky Gaussian Processescited by 2
  26. 2023Accelerating Non-Conjugate Gaussian Processes By Trading Off Computation For Uncertaintycited by 2
  27. 2023Gaussian Process Regression under Computational and Epistemic Misspecificationcited by 2
  28. 2024Scaling up Probabilistic PDE Simulators with Structured Volumetric Informationcited by 2
  29. 2024Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equationscited by 2
  30. 2025Online Conformal Probabilistic Numerics via Adaptive Edge-Cloud Offloadingcited by 1
  31. 2025Physics-based deep kernel learning for parameter estimation in high dimensional PDEscited by 1
  32. 2026Bayesian Quadraturecited by 1
  33. 2021A Probabilistically Motivated Learning Rate Adaptation for Stochastic Optimizationcited by 0
  34. 2021Black Box Probabilistic Numericscited by 0
  35. 2023Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inferencecited by 0
  36. 2023Deep Bayesian Reinforcement Learning for Spacecraft Proximity Maneuvers and Dockingcited by 0
  37. 2025An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equationcited by 0
  38. 2025Scalable h-adaptive probabilistic solver for time-independent and time-dependent systemscited by 0

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