# Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/1605.07811 · HTML: https://mlsanity.com/paper/1605.07811/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2016.

## Built on by (most cited first, 3 of 3)

- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [The Inverse of Exact Renormalization Group Flows as Statistical Inference](https://mlsanity.com/paper/2212.11379/) (2022)
- [Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operators](https://mlsanity.com/paper/2601.06462/) (2026)

## Abstract (arXiv)

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse problems whilst accounting, in a statistically principled way, for the impact of discretisation error due to numerical solution of the PDE. In particular, the approach confers robustness to failure of the numerical PDE solver, with statistical inferences driven to be more conservative in the presence of substantial discretisation error. Going further, the problem of choosing a PDE solver is cast as a problem in the Bayesian design of experiments, where the aim is to minimise the impact of solver error on statistical inferences; here the challenge of non-linear PDEs is also considered. The method is applied to parameter inference problems in which discretisation error in non-negligible and must be accounted for in order to reach conclusions that are statistically valid.