Probabilistic ODE Solvers with Runge-Kutta Means

TL;DR

This paper introduces a probabilistic framework for ODE solvers that aligns with Runge-Kutta methods, providing richer uncertainty quantification while maintaining their accuracy and efficiency.

Contribution

It develops a family of probabilistic ODE solvers with posterior means matching Runge-Kutta outputs, offering probabilistic insights and improved uncertainty estimates.

Findings

Posterior means exactly match Runge-Kutta solutions.

The probabilistic approach provides a richer output with uncertainty quantification.

The method maintains low computational cost similar to classical solvers.

Abstract

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.