Abstract
Complex systems manifest a small number of instabilities and bifurcations that are canonical in nature, resulting in universal pattern forming characteristics as a function of some parametric dependence. Such parametric instabilities are mathematically characterized by their universal unfoldings, or normal form dynamics, whereby a parsimonious model can be used to represent the dynamics. Although center manifold theory guarantees the existence of such lowdimensional normal forms, finding them has remained a long standing challenge. In this work, we introduce deep learning autoencoders to discover coordinate transformations that capture the underlying parametric dependence of a dynamical system in terms of its canonical normal form, allowing for a simple representation of the parametric dependence and bifurcation structure. The autoencoder constrains the latent variable to adhere to a given normal form, thus allowing it to learn the appropriate coordinate transformation. We demonstrate the method on a number of example problems, showing that it can capture a diverse set of normal forms associated with Hopf, pitchfork, transcritical and/or saddle node bifurcations. This method shows how normal forms can be leveraged as canonical and universal building blocks in deep learning approaches for model discovery and reducedorder modeling.
Original language  English 

Publisher  ArXiv.org 
Number of pages  18 
Publication status  Published  9 Jun 2021 
Keywords
 cs.LG
 math.DS
 37G05
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Normal Form Autoencoder, data associated with the publication: ‘Learning normal form autoencoders for datadriven discovery of universal, parameterdependent governing equations’
Kalia, M. (Creator), Meijer, H. G. E. (Creator), Brunton, S. L. (Creator), Kutz, J. N. (Creator) & Brune, C. (Creator), 4TU.Centre for Research Data, 18 Jun 2021
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