The computational bottleneck of building parametric surrogate models is mainly associated with the memory-intensive offline phase, where repeated solutions of physical (possibly large-scale, coupled, and nonlinear) problems are required. This cost depends on the spatial resolution of the employed models and on the dimensionality of the parametric space to be explored. This talk presents two strategies to address this computational bottleneck by reducing the spatial and parametric complexity of the problems. To reduce the number of coupled degrees of freedom in space, local reduced order models (ROMs) are devised using domain decomposition techniques within the proper generalised decomposition (PGD) framework [1,2]. The offline stage is thus optimised by employing the encapsulated PGD to construct independent, physics-based surrogate models in each subdomain, with parametrised Dirichlet boundary conditions at the interfaces. In the online phase, the local surrogate models are then evaluated for a new set of parameters by interpolating the PGD-ROMs without solving any additional reduced problem. Finally, they are coupled in real-time by imposing appropriate compatibility conditions at the interfaces. In order to handle the challenge of populating datasets for high-dimensional problems with representative snapshots, data augmentation techniques are proposed for a posteriori ROMs based on the proper orthogonal decomposition (POD) [3]. In this context, the cost of the offline phase is reduced by manufacturing artificial snapshots from a limited number of full-order solutions (data), while imposing conservation of mass and momentum (physics) by construction. The resulting enriched datasets enhance POD expressiveness, significantly improving the accuracy of the surrogates at a fraction of the cost of additional full-order solves. Numerical examples of parametric Stokes, Stokes-Darcy, and incompressible Navier-Stokes flows will be presented. The results showcase the capability of the proposed techniques to reduce the offline cost and enable more affordable surrogate models, without compromising their accuracy and expressiveness. [1] https://doi.org/10.1016/j.cma.2023.116484 [2] https://doi.org/10.48550/arXiv.2507.06861 [3] https://doi.org/10.1002/nme.7624