# A probabilistic model for the numerical solution of initial value problems

arXiv: https://arxiv.org/abs/1610.05261 · HTML: https://mlsanity.com/paper/1610.05261/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2016.

## Built on by (most cited first, 30 of 32)

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- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements](https://mlsanity.com/paper/2410.10530/) (2024)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)
- [Unlocking Out-of-Distribution Generalization in Dynamics through Physics-Guided Augmentation](https://mlsanity.com/paper/2510.24216/) (2025)
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## Abstract (arXiv)

Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a class of algorithms that formulate the solution to an IVP as inference on a latent path that is a draw from a Gaussian process probability measure (or equivalently, the solution of a linear stochastic differential equation). We then show that certain members of this class are connected precisely to generalized linear methods for ODEs, a number of Runge--Kutta methods, and Nordsieck methods. This probabilistic formulation of classic methods is valuable in two ways: analytically, it highlights implicit prior assumptions favoring certain approximate solutions to the IVP over others, and gives a precise meaning to the old observation that these methods act like filters. Practically, it endows the classic solvers with `docking points' for notions of uncertainty and prior information about the initial value, the value of the ODE itself, and the solution of the problem.