ABSTRACT:

We propose a new reduced-order modeling strategy for parametrized Darcy-type problems with linear constraints, such as mass conservation. Although based on standard neural networks within a supervised learning framework, the method is designed to strictly enforce the prescribed constraints.

The approach decomposes the PDE solution into a particular component that satisfies the constraint and a homogeneous component. The particular solution is efficiently constructed via a spanning-tree algorithm, while the homogeneous part is represented by a neural-network-generated potential projected onto the kernel of the constraint operator. We present three variants of the method, ranging from POD-based reduced spaces to more abstract constructions based on differential complexes, balancing computational efficiency with a solid mathematical foundation.

We validate the method through numerical experiments on porous media flow, including mixed-dimensional and nonlinear problems, demonstrating advantages over black-box models. Finally, we extend the framework to linear elasticity, enabling the approximation of stress, displacement, and rotation fields while preserving linear and angular momentum.

BIO:

Alessio Fumagalli is an Associate Professor of Numerical Analysis at the Department of Mathematics, Politecnico di Milano, where he served earlier as Assistant Professor (tenure track) at the same institution from 2019. He received his Ph.D. in Mathematical Models and Methods in Engineering from Politecnico di Milano in 2012, and had appointments as Postdoctoral Fellow at IFP Énergies Nouvelles in France, Postdoctoral Fellow at the University of Bergen, Norway, Politecnico di Milano and Visiting Professor at Politecnico di Torino.

His research focuses on mathematical modelling and non-standard numerical methods for flow and transport in fractured porous media, with particular emphasis on mixed-dimensional models and advanced discretization techniques such as XFEM and VEM.