# Probabilistic Numerics and Uncertainty in Computations

arXiv: https://arxiv.org/abs/1506.01326 · HTML: https://mlsanity.com/paper/1506.01326/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2015.

## Built on by (most cited first, 30 of 60)

- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Emulation of physical processes with Emukit](https://mlsanity.com/paper/2110.13293/) (2021)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [A Probabilistic Interpretation of Self-Paced Learning with Applications to Reinforcement Learning](https://mlsanity.com/paper/2102.13176/) (2021)
- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [High-Dimensional Gaussian Process Inference with Derivatives](https://mlsanity.com/paper/2102.07542/) (2021)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees](https://mlsanity.com/paper/2111.01721/) (2021)
- [Computation-Aware Kalman Filtering and Smoothing](https://mlsanity.com/paper/2405.08971/) (2024)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlations](https://mlsanity.com/paper/2203.03116/) (2022)
- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference](https://mlsanity.com/paper/2411.01036/) (2024)
- [Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunities](https://mlsanity.com/paper/2111.06223/) (2021)
- [Pick-and-Mix Information Operators for Probabilistic ODE Solvers](https://mlsanity.com/paper/2110.10770/) (2021)
- [Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach](https://mlsanity.com/paper/2306.05843/) (2023)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [BayeSQP: Bayesian Optimization through Sequential Quadratic Programming](https://mlsanity.com/paper/2602.03232/) (2026)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [Implicit Manifold Gaussian Process Regression](https://mlsanity.com/paper/2310.19390/) (2023)
- [Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation](https://mlsanity.com/paper/2405.13149/) (2024)
- [Advancing Trajectory Optimization with Approximate Inference: Exploration, Covariance Control and Adaptive Risk](https://mlsanity.com/paper/2103.06319/) (2021)
- [Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC](https://mlsanity.com/paper/2104.03942/) (2021)
- [Efficient Stochastic Optimal Control through Approximate Bayesian Input Inference](https://mlsanity.com/paper/2105.07693/) (2021)
- [PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2204.02583/) (2022)

## 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.