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

TL;DR

This paper introduces a neural network-based reduced order modeling approach for Darcy flow systems that guarantees exact satisfaction of mass conservation constraints, improving efficiency and interpretability in simulating parametrized PDEs.

Contribution

It presents a novel methodology combining neural networks with classical techniques to ensure linear constraints are satisfied exactly in reduced order models for Darcy flow systems.

Findings

The proposed models accurately enforce mass conservation in simulations.

Numerical experiments demonstrate improved efficiency over traditional black-box models.

The approach is applicable to complex porous media flow problems.

Abstract

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.