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Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

Tronarp, Filip LU ; Kersting, Hans ; Hennig, Philipp and Särkkä, Simo (2019) In Statistics and Computing 29(6). p.1297-1315
Abstract
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a nonlinear Bayesian filtering problem and all widely used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic... (More)
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a nonlinear Bayesian filtering problem and all widely used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the nonlinear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments. (Less)
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author
; ; and
publishing date
type
Contribution to journal
publication status
published
subject
in
Statistics and Computing
volume
29
issue
6
pages
1297 - 1315
publisher
Springer
external identifiers
  • scopus:85074030253
ISSN
0960-3174
DOI
10.1007/s11222-019-09900-1
language
English
LU publication?
no
id
64d92b3b-ff3a-41ea-950f-4ca8e7a64f2c
date added to LUP
2023-08-20 22:28:19
date last changed
2025-04-04 14:35:54
@article{64d92b3b-ff3a-41ea-950f-4ca8e7a64f2c,
  abstract     = {{We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a nonlinear Bayesian filtering problem and all widely used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the nonlinear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.}},
  author       = {{Tronarp, Filip and Kersting, Hans and Hennig, Philipp and Särkkä, Simo}},
  issn         = {{0960-3174}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{1297--1315}},
  publisher    = {{Springer}},
  series       = {{Statistics and Computing}},
  title        = {{Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective}},
  url          = {{http://dx.doi.org/10.1007/s11222-019-09900-1}},
  doi          = {{10.1007/s11222-019-09900-1}},
  volume       = {{29}},
  year         = {{2019}},
}