Conditional UDE

Repository accompanying the paper on conditional universal differential equations.

What is a conditional UDE?

Whereas universal differential equations do not directly accommodate the derivation of model components from data containing variability between samples, such as measurements on individuals in a population, the conditional UDE, combined with its specific training procedure, accounts for this through additional learnable inputs.

A cUDE is trained with a global neural network parameter set, and a set of conditional parameters that allow for explanation of the between-sample variability. This setup does then require a test set, where the neural network parameters are fixed and only the conditional parameters are estimated.

Structure of this repository

This repository contains the following folders:

• assets: illustrations in the Readme files.
• c-peptide: code model fits of the UDE and the final symbolic equations on the c-peptide data, as well as the computation of indices of beta-cell function.
• data: contains the data and the associated license used for this project.
• figures: contains the figures saved from the code.
• symbolic-regression: all code and raw result files from the symbolic regression run.
• source_data: all raw results from the code runs.
• src: contains the functions that are used in the code runs
• suppression: contains the code for the "suppression model" simulated example, which is used to illustrate the cUDE methodology.

Dependencies

Dependencies are split between Julia and Python.

Julia

Used packages and versions are documented in the Project.toml file. Julia's package manager pkg automatically takes care of installing the correct versions. To install the dependencies, open Julia in the repository path.

julia

Select the package manager by pressing ], and activate the environment.

pkg> activate .

Instantiate the environment to install the packages:

pkg> instantiate

Python

Poetry was used for package management. Packages are included in the pyproject.toml file.

Citation

When using the methodology/code from this repository, please cite:

• de Rooij, M., van Riel, N.A.W. & O’Donovan, S.D. Conditional universal differential equations capture population dynamics and interindividual variation in c-peptide production. npj Syst Biol Appl 11, 84 (2025). https://doi.org/10.1038/s41540-025-00570-6

BibTeX Entry

@article{de_rooij_conditional_2025,
	title = {Conditional universal differential equations capture population dynamics and interindividual variation in c-peptide production},
	volume = {11},
	copyright = {2025 The Author(s)},
	issn = {2056-7189},
	url = {https://www.nature.com/articles/s41540-025-00570-6},
	doi = {10.1038/s41540-025-00570-6},
	language = {en},
	number = {1},
	urldate = {2025-09-03},
	journal = {npj Systems Biology and Applications},
	author = {{de Rooij}, Max and {van Riel}, Natal A. W. and O’Donovan, Shauna D.},
	month = jul,
	year = {2025},
	note = {Publisher: Nature Publishing Group},
	keywords = {Biomedical engineering, Differential equations, Systems biology},
	pages = {84},
}

Data sources

The glucose, insulin, and c-peptide data during both the OGTT and the clamp experiments were originally taken from Okuno et al. (2013)¹, and also used by Ohashi et al. (2015)² and Ohashi et al. (2018)³. As an external dataset, OGTT data from 20 individuals was used from Fujita et al. (2023)⁴

Footnotes

1. Okuno, Y., Komada, H., Sakaguchi, K., Nakamura, T., Hashimoto, N., Hirota, Y., Ogawa, W., Seino, S.: Postprandial serum c-peptide to plasma glucose concentration ratio correlates with oral glucose tolerance test- and glucose clamp-based disposition indexes. Metabolism: Clinical and Experimental 62, 1470–1476 (2013) https://doi.org/10.1016/j.metabol.2013.05.022 ↩

2. Ohashi, K., Komada, H., Uda, S., Kubota, H., Iwaki, T., Fukuzawa, H., Komori, Y., Fujii, M., Toyoshima, Y., Sakaguchi, K., Ogawa, W., Kuroda, S.: Glucose homeostatic law: Insulin clearance predicts the progression of glucose intolerance in humans. PLOS ONE 10, 0143880 (2015) https://doi.org/10.1371/JOURNAL.PONE.0143880 ↩

3. Ohashi, K., Fujii, M., Uda, S., Kubota, H., Komada, H., Sakaguchi, K., Ogawa, W., Kuroda, S.: Increase in hepatic and decrease in peripheral insulin clearance characterize abnormal temporal patterns of serum insulin in diabetic subjects. npj Systems Biology and Applications 2018 4:1 4, 1–12 (2018) https://doi.org/10.1038/s41540-018-0051-6 ↩

4. Fujita, S., Karasawa, Y., Hironaka, K. I., Taguchi, Y. H., & Kuroda, S. (2023). Features extracted using tensor decomposition reflect the biological features of the temporal patterns of human blood multimodal metabolome. PLoS ONE, 18(2 February). https://doi.org/10.1371/journal.pone.0281594 ↩