The Neural Approximated Virtual Element Method for Elasticity Problems

TL;DR

This paper introduces a hybrid neural-approximated virtual element method for elasticity problems, combining classical numerical techniques with deep learning to simplify discretization and improve handling of non-linearities.

Contribution

It proposes a novel neural-approximated virtual element method that eliminates stabilization, enabling easier discretization of elasticity problems with non-linearities.

Findings

Effective in solving linear elasticity problems

Handles non-linear elasticity with improved simplicity

Demonstrates advantages over traditional methods

Abstract

We present the Neural Approximated Virtual Element Method to numerically solve elasticity problems. This hybrid technique combines classical concepts from the Finite Element Method and the Virtual Element Method with recent advances in deep neural networks. Specifically, it is a polygonal method in which the virtual basis functions are element-wise approximated by a neural network, eliminating the need for stabilization or projection operators typical of the standard virtual element method. We present the discrete formulation of the problem and provide numerical tests on both linear and non-linear elasticity problems, demonstrating the advantages of having a simple discretization, particularly in handling non-linearities.