# Probabilistic ODE Solvers with Runge-Kutta Means

arXiv: https://arxiv.org/abs/1406.2582 · HTML: https://mlsanity.com/paper/1406.2582/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2014.

## Built on by (most cited first, 22 of 22)

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- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
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- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Breaking the Discretization Barrier of Continuous Physics Simulation Learning](https://mlsanity.com/paper/2509.17955/) (2025)
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- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models](https://mlsanity.com/paper/2211.11103/) (2022)
- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (2024)
- [Numerically Robust Fixed-Point Smoothing Without State Augmentation](https://mlsanity.com/paper/2409.20004/) (2024)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Aligned Multi-Task Gaussian Process](https://mlsanity.com/paper/2110.15761/) (2021)
- [Advanced Global Wildfire Activity Modeling with Hierarchical Graph ODE](https://mlsanity.com/paper/2601.01501/) (2026)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [A probabilistic Taylor expansion with Gaussian processes](https://mlsanity.com/paper/2102.00877/) (2021)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Unlocking Out-of-Distribution Generalization in Dynamics through Physics-Guided Augmentation](https://mlsanity.com/paper/2510.24216/) (2025)

## Abstract (arXiv)

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.