Two continuous extensions of the Neural Approximated Virtual Element Method

TL;DR

This paper introduces two neural-based methods, B-NAVEM and P-NAVEM, that extend the Neural Approximated Virtual Element Method by ensuring global continuity and improving local basis function construction.

Contribution

The paper presents two novel neural variants of NAVEM that guarantee continuity and adapt basis functions using physics-informed and polynomial reproducibility techniques.

Findings

Both methods achieve competitive accuracy in numerical experiments.

B-NAVEM and P-NAVEM demonstrate different trade-offs in computational cost and memory usage.

The approaches effectively extend NAVEM with improved basis function continuity.

Abstract

We propose two globally continuous neural-based variants of the Neural Approximated Virtual Element Method (NAVEM), termed B-NAVEM and P-NAVEM. Both approaches construct local basis functions using pre-trained fully connected neural networks while ensuring exact continuity across adjacent mesh elements. B-NAVEM leverages a Physics-Informed Neural Network to approximately solve the local Laplace problem that defines the virtual element basis functions, whereas P-NAVEM directly enforces polynomial reproducibility via a tailored loss function, without requiring harmonicity within the element interior. Numerical experiments assess the methods in terms of computational cost, memory usage, and accuracy during both training and testing phases.