  A general, variational approach to derive low-order reduced systems is
presented. The approach is based on the concept of optimal parameterizing
manifold (OPM) that substitutes the more classical notions of invariant or slow
manifold when breakdown of "slaving" occurs, i.e. when the unresolved variables
cannot be expressed as an exact functional of the resolved ones anymore. The
OPM provides, within a given class of parameterizations of the unresolved
variables, the manifold that averages out optimally these variables as
conditioned on the resolved ones.
  The class of parameterizations retained here is that of continuous
deformations of parameterizations rigorously valid near the onset of
instability. These deformations are produced through integration of auxiliary
backward-forward (BF) systems built from the model's equations and lead to
analytic formulas for parameterizations. In this modus operandi, the backward
integration time is the key parameter to select per scale/variable to
parameterize in order to derive the relevant parameterizations which are doomed
to be no longer exact, away from instability onset, due to breakdown of slaving
typically encountered e.g. for chaotic regimes. The selection criterion is then
made through data-informed minimization of a least-square parameterization
defect. It is thus shown, through optimization of the backward integration time
per scale/variable to parameterize, that skilled OPM reduced systems can be
derived for predicting with accuracy higher-order critical transitions or
catastrophic tipping phenomena, while training our parameterization formulas
for regimes prior to these transitions take place.
