  In this work, we address parametric non-stationary fluid dynamics problems
within a model order reduction setting based on domain decomposition. Starting
from the optimisation-based domain decomposition approach, we derive an optimal
control problem, for which we present a convergence analysis in the case of
non-stationary incompressible Navier-Stokes equations. We discretize the
problem with the finite element method and we compare different model order
reduction techniques: POD-Galerkin and a non-intrusive neural network
procedure. We show that the classical POD-Galerkin is more robust and accurate
also in transient areas, while the neural network can obtain simulations very
quickly though being less precise in the presence of discontinuities in time or
parameter domain. We test the proposed methodologies on two fluid dynamics
benchmarks with physical parameters and time dependency: the non-stationary
backward-facing step and lid-driven cavity flow.
