# Black Box Probabilistic Numerics

arXiv: https://arxiv.org/abs/2106.13718 · HTML: https://mlsanity.com/paper/2106.13718/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 74 references that are in the map.
- First posted on arXiv 15 Jun 2021.

## Builds on (most cited first, 30 of 30)

- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Probabilistic Line Searches for Stochastic Optimization](https://mlsanity.com/paper/1502.02846/) (2015)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Calibrated Adaptive Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.08202/) (2020)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions](https://mlsanity.com/paper/2001.10965/) (2020)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Convergence Rates of Gaussian ODE Filters](https://mlsanity.com/paper/1807.09737/) (2018)
- [Probabilistic Interpretation of Linear Solvers](https://mlsanity.com/paper/1402.2058/) (2014)
- [Improving Quadrature for Constrained Integrands](https://mlsanity.com/paper/1802.04782/) (2018)
- [Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics](https://mlsanity.com/paper/1306.0308/) (2013)
- [A Bayes-Sard Cubature Method](https://mlsanity.com/paper/1804.03016/) (2018)
- [Probabilistic Linear Solvers for Machine Learning](https://mlsanity.com/paper/2010.09691/) (2020)
- [Active Uncertainty Calibration in Bayesian ODE Solvers](https://mlsanity.com/paper/1605.03364/) (2016)
- [Symmetry Exploits for Bayesian Cubature Methods](https://mlsanity.com/paper/1809.10227/) (2018)
- [Probabilistic Iterative Methods for Linear Systems](https://mlsanity.com/paper/2012.12615/) (2020)
- [Probabilistic Linear Multistep Methods](https://mlsanity.com/paper/1610.08417/) (2016)
- [Implicit Probabilistic Integrators for ODEs](https://mlsanity.com/paper/1805.07970/) (2018)
- [On the Bayesian Solution of Differential Equations](https://mlsanity.com/paper/1805.07109/) (2018)
- [A Fourier State Space Model for Bayesian ODE Filters](https://mlsanity.com/paper/2007.09118/) (2020)

## Abstract (arXiv)

Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson's deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems-a setting for which no probabilistic numerical methods have yet been developed.