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.

翻译：我们提出了一种新的降阶建模策略，用于处理具有线性约束的参数化偏微分方程（PDE），特别是由质量守恒给出约束的达西流系统。该方法采用经典神经网络架构和监督学习，但其构建方式能确保得到的降阶模型（ROM）严格满足线性约束。该过程将PDE解分解为满足约束的特解和齐次解：通过将神经网络模型生成的合适势函数映射到约束算子的零空间来近似齐次解；针对特解，我们提出了一种高效的生成树算法。基于这一范式，我们提出了三种遵循该方法的方案，通过探索势空间的不同选择来实现——从基于本征正交分解（POD）的经验空间，到基于微分复形的抽象空间。所有方案都将计算效率与严格的数学解释相结合，从而保证模型输出的可解释性。为展示所提策略的有效性并强调其相对于传统黑箱方法的优势，我们针对多孔介质中的流体流动开展了一系列数值实验，涵盖混合维度问题和非线性系统。本研究为模型降阶领域的深入探索与发展奠定了基础，有望在地球计算科学及相关领域解锁新的能力与解决方案。