: This paper deals with the development of a reduced order model (ROM) which could be used as an efficient tool for the reconstruction of the unsteady blood flow patterns in cardiovascular applications. The methodology relies on proper orthogonal decomposition to compute basis functions, combined with a Galerkin projection to compute the reduced coefficients. The main novelty of this work lies in the extension of the lifting function method, which typically is adopted for treating nonhomogeneous inlet velocity boundary conditions, to the handling of nonhomogeneous outlet boundary conditions for pressure, representing a very delicate point in numerical simulations of cardiovascular systems. Moreover, we incorporate a properly trained neural network in the ROM framework to approximate the mapping from time parameter to outflow pressure, which in the most general case is not available in closed form. We define our approach as "hybrid", because it merges equation-based elements with purely data-driven ones. The full order model (FOM) is related to a finite volume method which is employed for the discretization of unsteady Navier-Stokes equations while a two-element Windkessel model is adopted to enforce a reliable estimation of outflow pressure. Numerical results, firstly related to a 3D idealized blood vessel and then to a 3D patient-specific aortic arch, demonstrate that our ROM is able to accurately approximate the FOM with a significant reduction in computational cost.
