A trajectory-based loss function to learn missing terms in bifurcating dynamical systems

Please use this identifier to cite or link to this item:
Open Access logo originally created by the Public Library of Science (PLoS)
Title: A trajectory-based loss function to learn missing terms in bifurcating dynamical systems
Authors: Vortmeyer-Kley, Rahel
Nieters, Pascal
Pipa, Gordon
ORCID of the author: https://orcid.org/0000-0002-3416-2652
Abstract: Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.
Citations: Vortmeyer-Kley, R., Nieters, P. & Pipa, G: A trajectory-based loss function to learn missing terms in bifurcating dynamical systems. Sci Rep 11, 20394 (2021).
URL: https://doi.org/10.48693/230
Subject Keywords: Computational science; Nonlinear phenomena
Issue Date: 14-Oct-2021
License name: Attribution 4.0 International
License url: http://creativecommons.org/licenses/by/4.0/
Type of publication: Einzelbeitrag in einer wissenschaftlichen Zeitschrift [Article]
Appears in Collections:FB08 - Hochschulschriften

Files in This Item:
File Description SizeFormat 
SciReports_Vortmeyer-Kley_etal_2021.pdfArticle2,68 MBAdobe PDF

This item is licensed under a Creative Commons License Creative Commons