  This work introduces a non-intrusive model reduction approach for learning
reduced models from partially observed state trajectories of high-dimensional
dynamical systems. The proposed approach compensates for the loss of
information due to the partially observed states by constructing non-Markovian
reduced models that make future-state predictions based on a history of reduced
states, in contrast to traditional Markovian reduced models that rely on the
current reduced state alone to predict the next state. The core contributions
of this work are a data sampling scheme to sample partially observed states
from high-dimensional dynamical systems and a formulation of a regression
problem to fit the non-Markovian reduced terms to the sampled states. Under
certain conditions, the proposed approach recovers from data the very same
non-Markovian terms that one obtains with intrusive methods that require the
governing equations and discrete operators of the high-dimensional dynamical
system. Numerical results demonstrate that the proposed approach leads to
non-Markovian reduced models that are predictive far beyond the training
regime. Additionally, in the numerical experiments, the proposed approach
learns non-Markovian reduced models from trajectories with only 20% observed
state components that are about as accurate as traditional Markovian reduced
models fitted to trajectories with 99% observed components.
