Speaker
Alessio Fumagalli, Politecnico di Milano
Abstract
We propose a new strategy for reduced order modeling of parametrized Darcy-type problems with linear constraints, such as mass conservation. While our approach employs standard neural network architectures within a supervised learning framework, it is specifically designed so that the resulting Reduced Order Model strictly satisfies the imposed constraints.
The key idea is to decompose the PDE solution into two parts: a particular solution that enforces the constraint, and a homogeneous component. The particular solution is efficiently constructed using a spanning tree algorithm, whereas the homogeneous component is represented by a potential function generated through a neural network and projected onto the kernel of the constraint operator. Within this framework, we introduce three variants of the method, ranging from Proper Orthogonal Decomposition (POD)-based empirical spaces to more abstract constructions grounded in differential complexes. Each variant combines computational efficiency with a clear mathematical foundation, thereby enhancing the interpretability of the model outputs.
We validate the approach through a series of numerical experiments on flow in porous media, including mixed-dimensional and nonlinear problems, highlighting its advantages over conventional black-box methods. This work lays the foundation for further developments in model order reduction, with potential impact across computational geosciences and related fields. X Finally, we extend the same principles to linear elasticity, providing a framework to approximate stress, displacement, and rotation fields while guaranteeing conservation of linear and angular momentum.