Learning normal form autoencoders for data-driven discovery of universal, parameter-dependent governing equations

Manu Kalia, Steven L. Brunton, Hil G.E. Meijer, Christoph Brune, J. Nathan Kutz

Research output: Working paper

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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 un-foldings, or normal form dynamics, whereby a parsimonious model can be used to represent the dynamics. Although center manifold theory guarantees the existence of such low-dimensional 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 reduced-order modeling.
Original languageEnglish
PublisherarXiv.org
Number of pages18
Publication statusPublished - 9 Jun 2021

Keywords

  • cs.LG
  • math.DS
  • 37G05

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