Machine Learning–Enhanced Polytopal Finite Element Methods

Paola F. Antonietti (MOX, Politecnico di Milano)

Abstract: This talk explores the integration of Machine Learning (ML) techniques into polytopal Finite Element Methods for the numerical discretization of Partial Differential Equations, to improve accuracy, flexibility, and computational efficiency while addressing the challenges inherent in complex real-world applications, particularly in computational neuroscience.
We present several ML-driven enhancement strategies at different stages of the computational pipeline. In the first part of the talk, we introduce novel ML-based algorithms for mesh agglomeration and refinement in polytopal FEMs. Specifically, we propose novel algorithms based on Graph Neural Networks for efficiently constructing high-quality, agglomerated (coarse) grids that preserve geometric accuracy while significantly reducing computational costs. The key feature is the ability to jointly process the mesh’s connectivity graph and the model’s underlying physical properties. Such agglomerated grids also serve as the basis for constructing grid hierarchies in geometric multigrid solvers, leading to notable improvements in overall performance. In the second part of the talk, we introduce a novel deep learning framework to accelerate the convergence of Multigrid iterative solvers. The proposed approach predicts the optimal strong-connection parameter and selects the optimal smoother, thereby decreasing the overall time to solution.  All the proposed methods are validated across various differential problems and discretization schemes, including Virtual Element Methods and polytopal Discontinuous Galerkin methods.
Finally, we illustrate the practical performance of the proposed approaches in an application in computational neuroscience: the simulation of neurodegenerative disease dynamics.