Calibrated Adaptive Probabilistic ODE Solvers

Nathanael Bosch, Philipp Hennig, Filip Tronarp
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3466-3474, 2021.

Abstract

Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver’s step-size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-bosch21a, title = { Calibrated Adaptive Probabilistic ODE Solvers }, author = {Bosch, Nathanael and Hennig, Philipp and Tronarp, Filip}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {3466--3474}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/bosch21a/bosch21a.pdf}, url = {https://proceedings.mlr.press/v130/bosch21a.html}, abstract = { Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver’s step-size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method. } }
Endnote
%0 Conference Paper %T Calibrated Adaptive Probabilistic ODE Solvers %A Nathanael Bosch %A Philipp Hennig %A Filip Tronarp %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-bosch21a %I PMLR %P 3466--3474 %U https://proceedings.mlr.press/v130/bosch21a.html %V 130 %X Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver’s step-size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.
APA
Bosch, N., Hennig, P. & Tronarp, F.. (2021). Calibrated Adaptive Probabilistic ODE Solvers . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3466-3474 Available from https://proceedings.mlr.press/v130/bosch21a.html.

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