  We present a new surrogate modeling technique for efficient approximation of
input-output maps governed by parametrized PDEs. The model is hierarchical as
it is built on a full order model (FOM), reduced order model (ROM) and
machine-learning (ML) model chain. The model is adaptive in the sense that the
ROM and ML model are adapted on-the-fly during a sequence of parametric
requests to the model. To allow for a certification of the model hierarchy, as
well as to control the adaptation process, we employ rigorous a posteriori
error estimates for the ROM and ML models. In particular, we provide an example
of an ML-based model that allows for rigorous analytical quality statements. We
demonstrate the efficiency of the modeling chain on a Monte Carlo and a
parameter-optimization example. Here, the ROM is instantiated by Reduced Basis
Methods and the ML model is given by a neural network or a VKOGA kernel model.
