Parallel-in-Time Probabilistic Numerical ODE Solvers

TL;DR

This paper introduces a parallel-in-time probabilistic numerical solver for ODEs that leverages Bayesian filtering to process all time steps simultaneously, significantly reducing computational time.

Contribution

It develops a novel parallel-in-time probabilistic ODE solver based on Bayesian filtering, improving computational efficiency over sequential methods.

Findings

Reduces time complexity from linear to logarithmic in the number of steps.

Demonstrates effectiveness on various ODEs.

Outperforms classic and existing probabilistic solvers.

Abstract

Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.