  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.
