  We propose a new reduced order modeling strategy for tackling parametrized
Partial Differential Equations (PDEs) with linear constraints, in particular
Darcy flow systems in which the constraint is given by mass conservation. Our
approach employs classical neural network architectures and supervised
learning, but it is constructed in such a way that the resulting Reduced Order
Model (ROM) is guaranteed to satisfy the linear constraints exactly. The
procedure is based on a splitting of the PDE solution into a particular
solution satisfying the constraint and a homogenous solution. The homogeneous
solution is approximated by mapping a suitable potential function, generated by
a neural network model, onto the kernel of the constraint operator; for the
particular solution, instead, we propose an efficient spanning tree algorithm.
Starting from this paradigm, we present three approaches that follow this
methodology, obtained by exploring different choices of the potential spaces:
from empirical ones, derived via Proper Orthogonal Decomposition (POD), to more
abstract ones based on differential complexes. All proposed approaches combine
computational efficiency with rigorous mathematical interpretation, thus
guaranteeing the explainability of the model outputs. To demonstrate the
efficacy of the proposed strategies and to emphasize their advantages over
vanilla black-box approaches, we present a series of numerical experiments on
fluid flows in porous media, ranging from mixed-dimensional problems to
nonlinear systems. This research lays the foundation for further exploration
and development in the realm of model order reduction, potentially unlocking
new capabilities and solutions in computational geosciences and beyond.
