# Convergence Rates of Gaussian ODE Filters

arXiv: https://arxiv.org/abs/1807.09737 · HTML: https://mlsanity.com/paper/1807.09737/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2018.

## Built on by (most cited first, 14 of 14)

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- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [Parallel-in-Time Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2310.01145/) (2023)
- [Linear-Time Probabilistic Solutions of Boundary Value Problems](https://mlsanity.com/paper/2106.07761/) (2021)
- [Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations](https://mlsanity.com/paper/2402.12231/) (2024)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)
- [Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs](https://mlsanity.com/paper/2606.08203/) (2026)

## Abstract (arXiv)

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss--Markov process boldsymbolX, which is then iteratively conditioned on information about dotx. This article establishes worst-case local convergence rates of order q+1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of q=1 and an integrated Brownian motion prior, and analyses how inaccurate information on dotx coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to q ∈ 2,3,dots.