Pick-and-Mix Information Operators for Probabilistic ODE Solvers

Nathanael Bosch, Filip Tronarp, Philipp Hennig
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:10015-10027, 2022.

Abstract

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-bosch22a, title = { Pick-and-Mix Information Operators for Probabilistic ODE Solvers }, author = {Bosch, Nathanael and Tronarp, Filip and Hennig, Philipp}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {10015--10027}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/bosch22a/bosch22a.pdf}, url = {https://proceedings.mlr.press/v151/bosch22a.html}, abstract = { Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions. } }
Endnote
%0 Conference Paper %T Pick-and-Mix Information Operators for Probabilistic ODE Solvers %A Nathanael Bosch %A Filip Tronarp %A Philipp Hennig %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-bosch22a %I PMLR %P 10015--10027 %U https://proceedings.mlr.press/v151/bosch22a.html %V 151 %X Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.
APA
Bosch, N., Tronarp, F. & Hennig, P.. (2022). Pick-and-Mix Information Operators for Probabilistic ODE Solvers . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:10015-10027 Available from https://proceedings.mlr.press/v151/bosch22a.html.

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