  Regressions created from experimental or simulated data enable the
construction of metamodels, widely used in a variety of engineering
applications. Many engineering problems involve multi-parametric physics whose
corresponding multi-parametric solutions can be viewed as a sort of
computational vademecum that, once computed offline, can be then used in a
variety of real-time engineering applications including optimization, inverse
analysis, uncertainty propagation or simulation based control. Sometimes, these
multi-parametric problems can be solved by using advanced model order reduction
-- MOR -- techniques. However, when the solution of these multi-parametric
problems becomes cumbersome, one possibility consists in solving the problem
for a sample of the parametric values, and then creating a regression from all
the computed solutions, to finally infer the solution for any choice of the
problem parameters. However, addressing high-dimensionality at the low data
limit, ensuring accuracy and avoiding overfitting constitutes a difficult
challenge. The present paper aims at proposing and discussing different
PGD-based advanced regressions enabling the just referred features.
