Initial Value Problem Uncertainty Propagation

Mathias Van Gompel, Tom Colemont, Tjonnie G.-F. Li, Johan Suykens, Frederik De Ceuster
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:198-211, 2026.

Abstract

Probabilistic ODE solvers quantify numerical uncertainty by returning a posterior distribution over the solution, rather than only a point estimate. However, these methods typically assume that the initial value problem (IVP) itself is deterministic and fully known. When initial conditions or parameters are uncertain, increasing the numerical accuracy during the solve should not reduce uncertainty about the IVP itself. Standard probabilistic ODE solvers do not distinguish clearly between numerical uncertainty and IVP uncertainty, and may therefore contract the latter inappropriately. Recent work addressed this issue by combining filtering-based probabilistic ODE solvers with numerical quadrature to marginalise correctly over the IVP uncertainty. We extend this work by formulating this outer marginalisation problem in a Bayesian quadrature framework, allowing uncertainty from the quadrature approximation itself to be quantified alongside propagated IVP uncertainty and conditional solver uncertainty. In addition, for Gaussian uncertainty in the initial conditions or parameters, we derive closed-form recursions that can be incorporated directly into filtering and smoothing methods, thereby avoiding numerical quadrature altogether.

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-van-gompel26a, title = {Initial Value Problem Uncertainty Propagation}, author = {Van Gompel, Mathias and Colemont, Tom and Li, Tjonnie G.-F. and Suykens, Johan and De Ceuster, Frederik}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {198--211}, 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/van-gompel26a/van-gompel26a.pdf}, url = {https://proceedings.mlr.press/v341/van-gompel26a.html}, abstract = {Probabilistic ODE solvers quantify numerical uncertainty by returning a posterior distribution over the solution, rather than only a point estimate. However, these methods typically assume that the initial value problem (IVP) itself is deterministic and fully known. When initial conditions or parameters are uncertain, increasing the numerical accuracy during the solve should not reduce uncertainty about the IVP itself. Standard probabilistic ODE solvers do not distinguish clearly between numerical uncertainty and IVP uncertainty, and may therefore contract the latter inappropriately. Recent work addressed this issue by combining filtering-based probabilistic ODE solvers with numerical quadrature to marginalise correctly over the IVP uncertainty. We extend this work by formulating this outer marginalisation problem in a Bayesian quadrature framework, allowing uncertainty from the quadrature approximation itself to be quantified alongside propagated IVP uncertainty and conditional solver uncertainty. In addition, for Gaussian uncertainty in the initial conditions or parameters, we derive closed-form recursions that can be incorporated directly into filtering and smoothing methods, thereby avoiding numerical quadrature altogether.} }
Endnote
%0 Conference Paper %T Initial Value Problem Uncertainty Propagation %A Mathias Van Gompel %A Tom Colemont %A Tjonnie G.-F. Li %A Johan Suykens %A Frederik De Ceuster %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-van-gompel26a %I PMLR %P 198--211 %U https://proceedings.mlr.press/v341/van-gompel26a.html %V 341 %X Probabilistic ODE solvers quantify numerical uncertainty by returning a posterior distribution over the solution, rather than only a point estimate. However, these methods typically assume that the initial value problem (IVP) itself is deterministic and fully known. When initial conditions or parameters are uncertain, increasing the numerical accuracy during the solve should not reduce uncertainty about the IVP itself. Standard probabilistic ODE solvers do not distinguish clearly between numerical uncertainty and IVP uncertainty, and may therefore contract the latter inappropriately. Recent work addressed this issue by combining filtering-based probabilistic ODE solvers with numerical quadrature to marginalise correctly over the IVP uncertainty. We extend this work by formulating this outer marginalisation problem in a Bayesian quadrature framework, allowing uncertainty from the quadrature approximation itself to be quantified alongside propagated IVP uncertainty and conditional solver uncertainty. In addition, for Gaussian uncertainty in the initial conditions or parameters, we derive closed-form recursions that can be incorporated directly into filtering and smoothing methods, thereby avoiding numerical quadrature altogether.
APA
Van Gompel, M., Colemont, T., Li, T.G., Suykens, J. & De Ceuster, F.. (2026). Initial Value Problem Uncertainty Propagation. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:198-211 Available from https://proceedings.mlr.press/v341/van-gompel26a.html.

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