# Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

arXiv: https://arxiv.org/abs/1306.0308 · HTML: https://mlsanity.com/paper/1306.0308/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2013.

## Built on by (most cited first, 11 of 11)

- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Parallel-in-Time Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2310.01145/) (2023)
- [Practical Bayesian Algorithm Execution via Posterior Sampling](https://mlsanity.com/paper/2410.20596/) (2024)
- [Linear-Time Probabilistic Solutions of Boundary Value Problems](https://mlsanity.com/paper/2106.07761/) (2021)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)

## Abstract (arXiv)

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.