Assessment of uncertainty quantification in universal differential equations
- Author
- Nina Schmid (UGent) , David Fernandes del Pozo (UGent) , Willem Waegeman (UGent) and Jan Hasenauer
- Organization
- Project
- Abstract
- Scientific machine learning is a new class of approaches that integrate physical knowledge and mechanistic models with data-driven techniques to uncover the governing equations of complex processes. Among the available approaches, universal differential equations (UDEs) combine prior knowledge in the form of mechanistic formulations with universal function approximators, such as neural networks. Integral to the efficacy of UDEs is the joint estimation of parameters for both the mechanistic formulations and the universal function approximators using empirical data. However, the robustness and applicability of these resultant models hinge upon the rigorous quantification of uncertainties associated with their parameters and predictive capabilities. In this work, we provide a formalization of uncertainty quantification (UQ) for UDEs and investigate key frequentist and Bayesian methods. By analyzing three synthetic examples of varying complexity, we evaluate the validity and efficiency of ensembles, variational inference and Markov-chain Monte Carlo sampling as epistemic UQ methods for UDEs.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'.
- Keywords
- uncertainty quantification, universal differential equations, scientific machine learning, SYSTEMS
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-01JRA2CTXCH8PQ5GZCD009HQGA
- MLA
- Schmid, Nina, et al. “Assessment of Uncertainty Quantification in Universal Differential Equations.” PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, vol. 383, no. 2293, 2025, pp. 1–14, doi:10.1098/rsta.2024.0444.
- APA
- Schmid, N., Fernandes del Pozo, D., Waegeman, W., & Hasenauer, J. (2025). Assessment of uncertainty quantification in universal differential equations. PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 383(2293), 1–14. https://doi.org/10.1098/rsta.2024.0444
- Chicago author-date
- Schmid, Nina, David Fernandes del Pozo, Willem Waegeman, and Jan Hasenauer. 2025. “Assessment of Uncertainty Quantification in Universal Differential Equations.” PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 383 (2293): 1–14. https://doi.org/10.1098/rsta.2024.0444.
- Chicago author-date (all authors)
- Schmid, Nina, David Fernandes del Pozo, Willem Waegeman, and Jan Hasenauer. 2025. “Assessment of Uncertainty Quantification in Universal Differential Equations.” PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 383 (2293): 1–14. doi:10.1098/rsta.2024.0444.
- Vancouver
- 1.Schmid N, Fernandes del Pozo D, Waegeman W, Hasenauer J. Assessment of uncertainty quantification in universal differential equations. PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES. 2025;383(2293):1–14.
- IEEE
- [1]N. Schmid, D. Fernandes del Pozo, W. Waegeman, and J. Hasenauer, “Assessment of uncertainty quantification in universal differential equations,” PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, vol. 383, no. 2293, pp. 1–14, 2025.
@article{01JRA2CTXCH8PQ5GZCD009HQGA,
abstract = {{Scientific machine learning is a new class of approaches that integrate physical knowledge and mechanistic models with data-driven techniques to uncover the governing equations of complex processes. Among the available approaches, universal differential equations (UDEs) combine prior knowledge in the form of mechanistic formulations with universal function approximators, such as neural networks. Integral to the efficacy of UDEs is the joint estimation of parameters for both the mechanistic formulations and the universal function approximators using empirical data. However, the robustness and applicability of these resultant models hinge upon the rigorous quantification of uncertainties associated with their parameters and predictive capabilities. In this work, we provide a formalization of uncertainty quantification (UQ) for UDEs and investigate key frequentist and Bayesian methods. By analyzing three synthetic examples of varying complexity, we evaluate the validity and efficiency of ensembles, variational inference and Markov-chain Monte Carlo sampling as epistemic UQ methods for UDEs.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'.}},
author = {{Schmid, Nina and Fernandes del Pozo, David and Waegeman, Willem and Hasenauer, Jan}},
issn = {{1364-503X}},
journal = {{PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}},
keywords = {{uncertainty quantification,universal differential equations,scientific machine learning,SYSTEMS}},
language = {{eng}},
number = {{2293}},
pages = {{1--14}},
title = {{Assessment of uncertainty quantification in universal differential equations}},
url = {{http://doi.org/10.1098/rsta.2024.0444}},
volume = {{383}},
year = {{2025}},
}
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