Artificial neural network evaluation of geometric constants for polygonal domains

Abstract

We propose an approach based on Artificial Neural Networks (ANNs) to evaluate geometric constants relevant to the analysis and design of numerical schemes for partial differential equations. These constants play a central role, significantly influencing, for instance, a posteriori error estimates and the overall design of the computational strategy. Our technique leverages ANNs to learn the dependencies between these constants and a set of descriptive geometric features associated to polytopal mesh elements. The main computational costs are confined to data processing and training phases, which can be performed offline once and for all. This yields an effective tool for computing the constants, which we verify and show to be applicable across different scenarios, without substantial modifications - demonstrating its broader usability beyond the specific example considered.