# Calibrated Adaptive Probabilistic ODE Solvers

arXiv: https://arxiv.org/abs/2012.08202 · HTML: https://mlsanity.com/paper/2012.08202/

## Key facts

- Cited by 21 papers in the map (data through 13 Oct 2026).
- First posted on arXiv 15 Dec 2020.

## Built on by (most cited first, 21 of 21)

- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [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)
- [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)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [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)
- [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)
- [Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Time](https://mlsanity.com/paper/2404.12923/) (2024)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [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)
- [Bayesian Surrogate Training on Multiple Data Sources: A Hybrid Modeling Strategy](https://mlsanity.com/paper/2412.11875/) (2024)
- [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)

Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.