Artificial neural network evaluation of geometric constants for polygonal domains

TL;DR

This paper introduces a neural network-based method to efficiently estimate geometric constants for polygonal domains, aiding numerical PDE analysis and design with broad applicability and offline training.

Contribution

It presents a novel ANN approach to learn geometric constants from mesh features, reducing computational costs and enabling versatile application across various scenarios.

Findings

ANN accurately predicts geometric constants

Method reduces offline computational costs

Applicable to diverse polygonal domain scenarios

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.