  We consider machine-learning of time-dependent quantities of interest derived
from solution trajectories of parabolic partial differential equations. For
large-scale or long-time integration scenarios, where using a full order model
(FOM) to generate sufficient training data is computationally prohibitive, we
propose an adaptive hierarchy of intermediate Reduced Basis reduced order
models (ROM) to augment the FOM training data by certified ROM training data
required to fit a kernel model.
