Parallel-in-Time Probabilistic Numerical ODE Solvers

Nathanael Bosch; Adrien Corenflos; Fatemeh Yaghoobi; Filip Tronarp; Philipp  Hennig; Simo Särkkä

Parallel-in-Time Probabilistic Numerical ODE Solvers

Nathanael Bosch, Adrien Corenflos, Fatemeh Yaghoobi, Filip Tronarp, Philipp Hennig, Simo Särkkä

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

2 Sitaatiot (Scopus)

32 Lataukset (Pure)

Abstrakti

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.

Alkuperäiskieli	Englanti
Artikkeli	206
Sivut	1-27
Sivumäärä	27
Julkaisu	Journal of Machine Learning Research
Vuosikerta	25
Tila	Julkaistu - 24 heinäk. 2024
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Rahoitus

The authors gratefully acknowledge co-funding by the European Union (ERC, ANUBIS, 101123955. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them). Philipp Hennig is a member of the Machine Learning Cluster of Excellence, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC number 2064/1 – Project number 390727645; he also gratefully acknowledges the German Federal Ministry of Education and Research (BMBF) through the Tubingen AI Center (FKZ: 01IS18039A); and funds from the Ministry of Science, Research and Arts of the State of Baden-Wurttemberg. The authors would like to thank Research Council of Finland for funding. Filip Tronarp was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Nathanael Bosch. The authors are grateful to Nicholas Krämer for many valuable discussion and to Jonathan Schmidt for feedback on the manuscript,

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Parallel-in-Time Probabilistic Numerical ODE Solvers Final published version, 1,05 MBLisenssi: CC BY

https://www.jmlr.org/papers/v25/23-1261.htmlLisenssi: CC BY

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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 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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AU - Yaghoobi, Fatemeh

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AU - Hennig, Philipp

AU - Särkkä, Simo

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N2 - 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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KW - Ordinary differential equations

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Parallel-in-Time Probabilistic Numerical ODE Solvers

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