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Citation¤

Please consider citing Probdiffeq and its algorithms if it helps you in your research. Here are some concrete suggestions for how.

Essential citations¤

If you use Probdiffeq in your research, please cite:

Krämer, N. (2023). Implementing probabilistic numerical solvers for differential equations (Doctoral dissertation, Dissertation, Tübingen, Universität Tübingen, 2024).

Here is a BibTeX:

@phdthesis{kramer2024implementing,
  title={Implementing probabilistic numerical solvers for differential equations},
  author={Kr{\"a}mer, Peter Nicholas},
  year={2024},
  school={Universit{"a}t T{"u}bingen}
}
The PDF explains the mathematics and algorithms behind this library.
If there is one text to reference when acknowledging Probdiffeq, it is the PhD thesis above.

However, there are some additional references that are critical to this library:

Adaptive time-stepping: When using adaptive time-stepping, also cite the adaptive step-sizing paper:

Nicholas Krämer (2025). Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements. In Kanagawa, M., Cockayne, J., Gessner, A., & Hennig, P. (Eds.), Proceedings of the First International Conference on Probabilistic Numerics, 12–24. PMLR.

Here is a BibTeX:

@InProceedings{kramer2024adaptive,
  title = {Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements},
  author = {Kr{\"a}mer, Nicholas},
  booktitle = {Proceedings of the First International Conference on Probabilistic Numerics},
  pages = {12--24},
  year = {2025},
  editor = {Kanagawa, Motonobu and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp},
  volume = {271},
  series = {Proceedings of Machine Learning Research},
  publisher = {PMLR},
  url = {https://proceedings.mlr.press/v271/kramer25a.html}
}
Link to the paper: PDF.

Link to the experiments: Code for experiments.

Numerical implementations: If you use more than one or two Taylor coefficients in the state-space model, you're benefiting from numerically robust implementations of probabilistic solvers:

Nicholas Krämer & Philipp Hennig (2024). Stable implementation of probabilistic ODE solvers. Journal of Machine Learning Research, 25(111), 1–29.

Here is a BibTeX:

@article{kraemer2024stable,
  title={Stable implementation of probabilistic ODE solvers},
  author={Kraemer, Nicholas and Hennig, Philipp},
  journal={Journal of Machine Learning Research},
  volume={25},
  number={111},
  pages={1--29},
  year={2024}
}

Specific algorithms¤

Algorithms in Probdiffeq are based on multiple research papers. If you’re unsure which to cite, feel free to reach out. A (subjective, probdiffeq-centric) list of relevant work includes the following articles.

Numerical robustness and state-space model factorisations¤

  • Nicholas Krämer & Philipp Hennig (2024). Stable implementation of probabilistic ODE solvers. Journal of Machine Learning Research, 25(111), 1–29.

    Key insights: All suggestions made in this work are critical to numerical implementations of probabilistic solvers. They are implemented by Probdiffeq (and other libraries).

  • Nicholas Krämer, Nathanael Bosch, Jonathan Schmidt & Philipp Hennig (2022). Probabilistic ODE solutions in millions of dimensions. In ICML 2022, 11634–11649. PMLR.

    Key insights: Every time Probdiffeq uses state-space model factorisations, it follows the recommendations in this work.

Adaptive step-size selection (and calibration)¤

  • Michael Schober, Simo Särkkä & Philipp Hennig (2019). A probabilistic model for the numerical solution of initial value problems. Statistics and Computing, 29(1), 99–122.

    Key insights: This work is the first on calibration and adaptive step-size selection in state-space-model-based ODE solvers.

  • Nathanael Bosch, Philipp Hennig & Filip Tronarp (2021). Calibrated adaptive probabilistic ODE solvers. In AISTATS 2021, 3466–3474. PMLR.

    Key insights: This work describes calibration and adaptive step-size selection as we use it now.

  • Nicholas Krämer, Nathanael Bosch, Jonathan Schmidt & Philipp Hennig (2022). Probabilistic ODE solutions in millions of dimensions. In ICML 2022, 11634–11649. PMLR.

    Key insights: This work is a small extension of Bosch et al. (2021)'s calibration and error estimates to factorised state-space models.

  • Nicholas Krämer (2025). Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements. In Kanagawa, M., Cockayne, J., Gessner, A., & Hennig, P. (Eds.), Proceedings of the First International Conference on Probabilistic Numerics, 12–24. PMLR.

    Key insights: Adaptive time-stepping with fixed-point smoothers makes memory requirements constant. Probdiffeq's time-stepping loop implements this paper.

Constraints, linearisation, and information operators¤

  • Tronarp, Filip, et al. "Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective." Statistics and Computing 29.6 (2019): 1297-1315.

    Key insight: As one of the foundational works on probabilistic solvers, it links ODE solvers to zeroth- and first-order linearisation in Gaussian filters.

  • Bosch, Nathanael, Filip Tronarp, and Philipp Hennig. "Pick-and-mix information operators for probabilistic ODE solvers." International Conference on Artificial Intelligence and Statistics. PMLR, 2022.

    Key insights: Encode, e.g. second-order dynamics, Hamiltonian preservation, or implicit differential equations directly in the constraints without transforming the problem into a first-order explicit ODE.

Parameter estimation¤

  • Kersting, H., Krämer, N., Schiegg, M., Daniel, C., Tiemann, M., & Hennig, P. (2020, November). Differentiable likelihoods for fast inversion of likelihood-free dynamical systems. In International Conference on Machine Learning (pp. 5198-5208). PMLR.

    Key insight: The first work on using the likelihood of observational data under a posterior distribution given by the probabilistic ODE solution.

  • Tronarp, Filip, Nathanael Bosch, and Philipp Hennig. "Fenrir: Physics-enhanced regression for initial value problems." International Conference on Machine Learning. PMLR, 2022.

    Key insight: The formulation of the likelihood of the observational data as we use it now.

  • Beck, J., Bosch, N., Deistler, M., Kadhim, K. L., Macke, J. H., Hennig, P., & Berens, P. (2024, July). Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations. In International Conference on Machine Learning (pp. 3305-3326). PMLR.

    Key insight: An improved algorithm for parameter estimation using the above likelihood formulation based on diffusion tempering (see the tutorial).

Prior distributions¤

  • Schober, M., Duvenaud, D., & Hennig, P. (2014). Probabilistic ODE solvers with Runge-Kutta means. Advances in neural information processing systems, 27.

    Key insights: Use Gauss--Markov processes, specifically, high-order integrated Wiener processes, to replicate the efficiency of non-probabilistic ODE solvers.

  • Kersting, H., Sullivan, T. J., & Hennig, P. (2020). Convergence rates of Gaussian ODE filters. Statistics and computing, 30(6), 1791-1816.

    Key insights: One of the first works that mentions integrated Ornstein-Uhlenbeck priors in the context of ODE solvers.

  • Bosch, Nathanael, Philipp Hennig, and Filip Tronarp. "Probabilistic exponential integrators." Advances in Neural Information Processing Systems 36 (2023): 40450-40467.

    Key insights: Replicate the behaviour of exponential integrators by choosing priors different to integrated Wiener processes.