In this work, we introduce a novel collocation-based Model Order Reduction strategy, called collocation Model Order Reduction (cMOR), proposed as an alternative to classical projection-based Model Order Reduction (pMOR). Unlike pMOR, which relies on Galerkin or Petrov–Galerkin projection onto a reduced space, cMOR determines the reduced solution by solving the governing equations at a strategically selected set of collocation points, identified through hyper-reduction techniques. The method preserves the typical two-stage structure of MOR: an offline phase, where a Reduced Basis (RB) is constructed from snapshots, and an online phase, where the High-Dimensional Model (HDM) is evaluated only on the empirical subset of points and the full solution is reconstructed via the RB.

The cMOR framework is analyzed theoretically in terms of stability and convergence, and it is applied within both linear reduced spaces and nonlinear approximation manifolds (NAM), the latter enhancing the ability to represent complex multiscale structures by embedding the solution on nonlinear manifolds rather than linear subspaces. Moreover, the method is implemented using advanced numerical technologies, including ADER schemes on unsteady Chimera grids, enabling efficient treatment of convection-dominated PDEs on evolving geometries. These applications highlight the robustness of cMOR in challenging scenarios—such as advection-dominated dynamics, moving multi-block grids, and high-dimensional parametric variations—where classical pMOR often suffers from instability or reduced accuracy due to the Kolmogorov N-width barrier.

Beyond traditional discretization frameworks, cMOR also demonstrates strong compatibility with varied computational settings, from numerical schemes to grid-based solvers and neural-network-assisted reduced models, thanks to its collocation-driven formulation. Numerical experiments across these diverse contexts confirm that cMOR achieves substantial computational savings while maintaining high accuracy, ensuring straightforward integration into existing simulation workflows and machine-learning-augmented pipelines.