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.

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