Probabilistic Numerics for Hamiltonian Dynamics

Frederik De Ceuster, Tom Colemont, Mathias Van Gompel, Tjonnie G.-F. Li
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:39-48, 2026.

Abstract

Many ordinary differential equations (ODEs) encountered in science originate from physical principles that contain substantially more structure than the ODE alone. In particular, Hamiltonian systems arise from variational principles, possess a symplectic structure, and exhibit conservation laws induced by symmetries. Standard probabilistic ODE solvers, that typically condition on the residual of the ODE, can overlook this additional physical information. In this paper, we focus on Hamiltonian dynamics, we revisit variational integrators and propose a probabilistic-numerical extension, based on a physics-informed prior. Furthermore, we discuss the symmetry-based Bayesian ODE framework of Wang et al. (2020), clarifying the role of integrability in Hamiltonian systems and the related Lie-algebra structure in defining Bayesian formulations

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-de-ceuster26a, title = {Probabilistic Numerics for {H}amiltonian Dynamics}, author = {De Ceuster, Frederik and Colemont, Tom and Van Gompel, Mathias and Li, Tjonnie G.-F.}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {39--48}, year = {2026}, editor = {Karvonen, Toni and Bosch, Nathanael and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp and Kouw, Wouter}, volume = {341}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v341/main/assets/de-ceuster26a/de-ceuster26a.pdf}, url = {https://proceedings.mlr.press/v341/de-ceuster26a.html}, abstract = {Many ordinary differential equations (ODEs) encountered in science originate from physical principles that contain substantially more structure than the ODE alone. In particular, Hamiltonian systems arise from variational principles, possess a symplectic structure, and exhibit conservation laws induced by symmetries. Standard probabilistic ODE solvers, that typically condition on the residual of the ODE, can overlook this additional physical information. In this paper, we focus on Hamiltonian dynamics, we revisit variational integrators and propose a probabilistic-numerical extension, based on a physics-informed prior. Furthermore, we discuss the symmetry-based Bayesian ODE framework of Wang et al. (2020), clarifying the role of integrability in Hamiltonian systems and the related Lie-algebra structure in defining Bayesian formulations} }
Endnote
%0 Conference Paper %T Probabilistic Numerics for Hamiltonian Dynamics %A Frederik De Ceuster %A Tom Colemont %A Mathias Van Gompel %A Tjonnie G.-F. Li %B Proceedings of the 2nd International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2026 %E Toni Karvonen %E Nathanael Bosch %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %E Wouter Kouw %F pmlr-v341-de-ceuster26a %I PMLR %P 39--48 %U https://proceedings.mlr.press/v341/de-ceuster26a.html %V 341 %X Many ordinary differential equations (ODEs) encountered in science originate from physical principles that contain substantially more structure than the ODE alone. In particular, Hamiltonian systems arise from variational principles, possess a symplectic structure, and exhibit conservation laws induced by symmetries. Standard probabilistic ODE solvers, that typically condition on the residual of the ODE, can overlook this additional physical information. In this paper, we focus on Hamiltonian dynamics, we revisit variational integrators and propose a probabilistic-numerical extension, based on a physics-informed prior. Furthermore, we discuss the symmetry-based Bayesian ODE framework of Wang et al. (2020), clarifying the role of integrability in Hamiltonian systems and the related Lie-algebra structure in defining Bayesian formulations
APA
De Ceuster, F., Colemont, T., Van Gompel, M. & Li, T.G.. (2026). Probabilistic Numerics for Hamiltonian Dynamics. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:39-48 Available from https://proceedings.mlr.press/v341/de-ceuster26a.html.

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