Bayesian Inference of Discretization Error Means in ODEs via Ensemble Kalman Filtering

Shoji Toyota, Yuto Miyatake
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:181-197, 2026.

Abstract

We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean—referred to as the discretization error mean—is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. It depends on a step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh–Nagumo model demonstrate the effectiveness of the proposed approach.

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-toyota26a, title = {{B}ayesian Inference of Discretization Error Means in {O}DEs via Ensemble {K}alman Filtering}, author = {Toyota, Shoji and Miyatake, Yuto}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {181--197}, 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/toyota26a/toyota26a.pdf}, url = {https://proceedings.mlr.press/v341/toyota26a.html}, abstract = {We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean—referred to as the discretization error mean—is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. It depends on a step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh–Nagumo model demonstrate the effectiveness of the proposed approach.} }
Endnote
%0 Conference Paper %T Bayesian Inference of Discretization Error Means in ODEs via Ensemble Kalman Filtering %A Shoji Toyota %A Yuto Miyatake %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-toyota26a %I PMLR %P 181--197 %U https://proceedings.mlr.press/v341/toyota26a.html %V 341 %X We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean—referred to as the discretization error mean—is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. It depends on a step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh–Nagumo model demonstrate the effectiveness of the proposed approach.
APA
Toyota, S. & Miyatake, Y.. (2026). Bayesian Inference of Discretization Error Means in ODEs via Ensemble Kalman Filtering. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:181-197 Available from https://proceedings.mlr.press/v341/toyota26a.html.

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