  This work introduces a method for learning low-dimensional models from data
of high-dimensional black-box dynamical systems. The novelty is that the
learned models are exactly the reduced models that are traditionally
constructed with model reduction techniques that require full knowledge of
governing equations and operators of the high-dimensional systems. Thus, the
learned models are guaranteed to inherit the well-studied properties of reduced
models from traditional model reduction. The key ingredient is a new data
sampling scheme to obtain re-projected trajectories of high-dimensional systems
that correspond to Markovian dynamics in low-dimensional subspaces. The exact
recovery of reduced models from these re-projected trajectories is guaranteed
pre-asymptotically under certain conditions for finite amounts of data and for
a large class of systems with polynomial nonlinear terms. Numerical results
demonstrate that the low-dimensional models learned with the proposed approach
match reduced models from traditional model reduction up to numerical errors in
practice. The numerical results further indicate that low-dimensional models
fitted to re-projected trajectories are predictive even in situations where
models fitted to trajectories without re-projection are inaccurate and
unstable.
