Parallel-in-Time Probabilistic Numerical ODE Solvers
(2024) In Journal of Machine Learning Research 25. p.1-27- 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... (More)
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 computational complexity 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.
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- author
- Bosch, Nathanael ; Corenflos, Adrien ; Yaghoobi, Fatemeh ; Tronarp, Filip LU ; Hennig, Philipp and Särkkä, Simo
- organization
- publishing date
- 2024
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Bayesian filtering, numerical analysis, ordinary differential equations, parallel-in-time methods, probabilistic numerics, smoothing
- in
- Journal of Machine Learning Research
- volume
- 25
- pages
- 27 pages
- publisher
- Microtome Publishing
- external identifiers
-
- scopus:105018575344
- ISSN
- 1532-4435
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: ©2024 Nathanael Bosch, Adrien Corenflos, Fatemeh Yaghoobi, Filip Tronarp, Philipp Hennig and Simo Särkkä.
- id
- f9588da4-1279-453b-8d9a-dda7f8a48ebd
- alternative location
- http://jmlr.org/papers/v25/23-1261.html
- date added to LUP
- 2026-01-27 14:34:22
- date last changed
- 2026-01-27 14:35:11
@article{f9588da4-1279-453b-8d9a-dda7f8a48ebd,
abstract = {{<p>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 computational complexity 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.</p>}},
author = {{Bosch, Nathanael and Corenflos, Adrien and Yaghoobi, Fatemeh and Tronarp, Filip and Hennig, Philipp and Särkkä, Simo}},
issn = {{1532-4435}},
keywords = {{Bayesian filtering; numerical analysis; ordinary differential equations; parallel-in-time methods; probabilistic numerics; smoothing}},
language = {{eng}},
pages = {{1--27}},
publisher = {{Microtome Publishing}},
series = {{Journal of Machine Learning Research}},
title = {{Parallel-in-Time Probabilistic Numerical ODE Solvers}},
url = {{http://jmlr.org/papers/v25/23-1261.html}},
volume = {{25}},
year = {{2024}},
}