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Licensed Unlicensed Requires Authentication Published by De Gruyter July 9, 2021

Collocation based training of neural ordinary differential equations

  • Elisabeth Roesch , Christopher Rackauckas and Michael P. H. Stumpf EMAIL logo


The predictive power of machine learning models often exceeds that of mechanistic modeling approaches. However, the interpretability of purely data-driven models, without any mechanistic basis is often complicated, and predictive power by itself can be a poor metric by which we might want to judge different methods. In this work, we focus on the relatively new modeling techniques of neural ordinary differential equations. We discuss how they relate to machine learning and mechanistic models, with the potential to narrow the gulf between these two frameworks: they constitute a class of hybrid model that integrates ideas from data-driven and dynamical systems approaches. Training neural ODEs as representations of dynamical systems data has its own specific demands, and we here propose a collocation scheme as a fast and efficient training strategy. This alleviates the need for costly ODE solvers. We illustrate the advantages that collocation approaches offer, as well as their robustness to qualitative features of a dynamical system, and the quantity and quality of observational data. We focus on systems that exemplify some of the hallmarks of complex dynamical systems encountered in systems biology, and we map out how these methods can be used in the analysis of mathematical models of cellular and physiological processes.

Corresponding author: Michael P. H. Stumpf, School of BioSciences, Biosciences 4, The University of Melbourne, Royal Parade, Parkville, VIC3052, Australia; and School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, VIC3010, Australia, E-mail:


We gratefully acknowledge discussions with members of the Theoretical Systems Biology group at University of Melbourne, Australia, and at Imperial College London, United Kingdom, as well as with the Julia community. The information, data, or work presented herein was funded in part by ARPA-E under award numbers DE-AR0001222 and DE-AR0001211, and NSF award number IIP-1938400. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


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Received: 2020-04-06
Revised: 2021-05-03
Accepted: 2021-05-04
Published Online: 2021-07-09

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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