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We propose a new reduced order modeling strategy for tackling parametrizedPartial Differential Equations (PDEs) with linear constraints, in particularDarcy flow systems in which the constraint is given by mass conservation. Ourapproach employs classical neural network architectures and supervisedlearning, but it is constructed in such a way that the resulting Reduced OrderModel (ROM) is guaranteed to satisfy the linear constraints exactly. Theprocedure is based on a splitting of the PDE solution into a particularsolution satisfying the constraint and a homogenous solution. The homogeneoussolution is approximated by mapping a suitable potential function, generated bya neural network model, onto the kernel of the constraint operator; for theparticular solution, instead, we propose an efficient spanning tree algorithm.Starting from this paradigm, we present three approaches that follow thismethodology, obtained by exploring different choices of the potential spaces:from empirical ones, derived via Proper Orthogonal Decomposition (POD), to moreabstract ones based on differential complexes. All proposed approaches combinecomputational efficiency with rigorous mathematical interpretation, thusguaranteeing the explainability of the model outputs. To demonstrate theefficacy of the proposed strategies and to emphasize their advantages overvanilla black-box approaches, we present a series of numerical experiments onfluid flows in porous media, ranging from mixed-dimensional problems tononlinear systems. This research lays the foundation for further explorationand development in the realm of model order reduction, potentially unlockingnew capabilities and solutions in computational geosciences and beyond.

Deep learning based reduced order modeling of Darcy flow systems with local mass conservation