Probabilistic Exponential Integrators
(2023) 37th Conference on Neural Information Processing Systems, NeurIPS 2023 In Advances in Neural Information Processing Systems 36.- Abstract
Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator-with the added benefit of providing a... (More)
Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator-with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.
(Less)
- author
- Bosch, Nathanael ; Hennig, Philipp and Tronarp, Filip LU
- organization
- publishing date
- 2023
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- host publication
- Advances in Neural Information Processing Systems
- series title
- Advances in Neural Information Processing Systems
- volume
- 36
- conference name
- 37th Conference on Neural Information Processing Systems, NeurIPS 2023
- conference location
- New Orleans, United States
- conference dates
- 2023-12-10 - 2023-12-16
- external identifiers
-
- scopus:85191163195
- ISSN
- 1049-5258
- language
- English
- LU publication?
- yes
- id
- 802c6690-78aa-4014-8cca-2a53b5fdd763
- date added to LUP
- 2024-05-07 10:32:52
- date last changed
- 2024-05-07 10:33:50
@inproceedings{802c6690-78aa-4014-8cca-2a53b5fdd763,
abstract = {{<p>Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator-with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.</p>}},
author = {{Bosch, Nathanael and Hennig, Philipp and Tronarp, Filip}},
booktitle = {{Advances in Neural Information Processing Systems}},
issn = {{1049-5258}},
language = {{eng}},
series = {{Advances in Neural Information Processing Systems}},
title = {{Probabilistic Exponential Integrators}},
volume = {{36}},
year = {{2023}},
}