# Bayesian ODE Solvers: The Maximum A Posteriori Estimate

arXiv: https://arxiv.org/abs/2004.00623 · HTML: https://mlsanity.com/paper/2004.00623/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2020.

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

- [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)
- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (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)
- [Learning Stochastic Dynamical Systems with Structured Noise](https://mlsanity.com/paper/2503.01077/) (2025)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)
- [Constructive Disintegration and Conditional Modes](https://mlsanity.com/paper/2508.00617/) (2025)
- [Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs](https://mlsanity.com/paper/2606.08203/) (2026)

## Abstract (arXiv)

It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a Gauss--Markov prior is used. In this paper the class of nu times differentiable linear time invariant Gauss--Markov priors is considered. A taxonomy of Gaussian estimators is established, with the maximum a posteriori estimate at the top of the hierarchy, which can be computed with the iterated extended Kalman smoother. The remaining three classes are termed explicit, semi-implicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness nu+1. Consequently, using methods from scattered data approximation and nonlinear analysis in Sobolev spaces, it is shown that the maximum a posteriori estimate converges to the true solution at a polynomial rate in the fill-distance (maximum step size) subject to mild conditions on the vector field. The methodology developed provides a novel and more natural approach to study the convergence of these estimators than classical methods of convergence analysis. The methods and theoretical results are demonstrated in numerical examples.