  While data-driven model reduction techniques are well-established for
linearizable mechanical systems, general approaches to reducing
non-linearizable systems with multiple coexisting steady states have been
unavailable. In this paper, we review such a data-driven nonlinear model
reduction methodology based on spectral submanifolds (SSMs). As input, this
approach takes observations of unforced nonlinear oscillations to construct
normal forms of the dynamics reduced to very low dimensional invariant
manifolds. These normal forms capture amplitude-dependent properties and are
accurate enough to provide predictions for non-linearizable system response
under the additions of external forcing. We illustrate these results on
examples from structural vibrations, featuring both synthetic and experimental
data.
