The lowest-order Neural Approximated Virtual Element Method on polygonal elements

TL;DR

This paper introduces a neural network-based approach to approximate virtual element basis functions on polygonal meshes, simplifying the method and improving performance especially on complex meshes with hanging nodes.

Contribution

It proposes a novel neural approximation technique for virtual element basis functions, reducing stabilization and projection issues in polygonal element methods.

Findings

Effective on general polygonal meshes

Improves accuracy over standard VEM

Handles meshes with hanging nodes

Abstract

The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here. This method employs a neural network to locally approximate the Virtual Element basis functions, thereby eliminating issues concerning stabilization and projection operators, which are the key components of the standard Virtual Element Method. We propose different training strategies for the neural network training, each correlated by the theoretical justification and with a different level of accuracy. Several numerical experiments are proposed to validate our procedure on general polygonal meshes and demonstrate the advantages of the proposed method across different problem formulations, particularly in cases where the heavy usage of projection and stabilization terms may represent challenges for the standard version of the method. Particular attention is reserved to triangular meshes with hanging nodes which assume a central role in many virtual element applications.