Deep Learning‐Based Reduced‐Order Modeling of Darcy‐Flow Systems With Local Mass Conservation

We propose a new reduced-order modeling (ROM) strategy for tackling parametrized 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 ROM is guaranteed to satisfy the linear constraints exactly. The procedure is based on a splitting of the 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: derived either via proper orthogonal decomposition (POD) or using the properties of a differential complex. To demonstrate the effectiveness of the proposed strategies and to emphasize their advantages over neural-network regression approaches, we present a series of numerical experiments, ranging from mixed-dimensional problems to nonlinear systems.