# Stable Implementation of Probabilistic ODE Solvers

arXiv: https://arxiv.org/abs/2012.10106 · HTML: https://mlsanity.com/paper/2012.10106/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2020.

## Built on by (most cited first, 19 of 19)

- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Pick-and-Mix Information Operators for Probabilistic ODE Solvers](https://mlsanity.com/paper/2110.10770/) (2021)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [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)
- [Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps](https://mlsanity.com/paper/2602.12624/) (2026)
- [Numerically Robust Fixed-Point Smoothing Without State Augmentation](https://mlsanity.com/paper/2409.20004/) (2024)
- [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)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [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)
- [Numerically robust Gaussian state estimation with singular observation noise](https://mlsanity.com/paper/2503.10279/) (2025)
- [Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems](https://mlsanity.com/paper/2508.09623/) (2025)
- [Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs](https://mlsanity.com/paper/2606.08203/) (2026)
- [Modified Bryson-Frazier Smoothing and Hyperparameter Learning for Temporal Gaussian Process Regression](https://mlsanity.com/paper/2608.17595/) (2026)

## Abstract (arXiv)

Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of theoretical analysis. However, these algorithms suffer from numerical instability when run at high order or with small step-sizes -- that is, exactly in the regime in which they achieve the highest accuracy. The present work proposes and examines a solution to this problem. It involves three components: accurate initialisation, a coordinate change preconditioner that makes numerical stability concerns step-size-independent, and square-root implementation. Using all three techniques enables numerical computation of probabilistic solutions of ODEs with algorithms of order up to 11, as demonstrated on a set of challenging test problems. The resulting rapid convergence is shown to be competitive to high-order, state-of-the-art, classical methods. As a consequence, a barrier between analysing probabilistic ODE solvers and applying them to interesting machine learning problems is effectively removed.