  We develop a methodology to construct low-dimensional predictive models from
data sets representing essentially nonlinear (or non-linearizable) dynamical
systems with a hyperbolic linear part that are subject to external forcing with
finitely many frequencies. Our data-driven, sparse, nonlinear models are
obtained as extended normal forms of the reduced dynamics on low-dimensional,
attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate
the power of data-driven SSM reduction on high-dimensional numerical data sets
and experimental measurements involving beam oscillations, vortex shedding and
sloshing in a water tank. We find that SSM reduction trained on unforced data
also predicts nonlinear response accurately under additional external forcing.
