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

Stefano Berrone, Moreno Pintore, Gioana Teora

TL;DR

The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here, employing 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.

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.

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

TL;DR

The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here, employing 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.

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.
Paper Structure (22 sections, 3 theorems, 68 equations, 12 figures, 2 tables)

This paper contains 22 sections, 3 theorems, 68 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Model Problem and The Virtual Element Method
  3. The Virtual Element Space
  4. The Virtual Element Discretization
  5. The Neural Approximated Virtual Element Method
  6. The Local Neural Approximated Virtual Element Space
  7. The NAVEM discretization and the Online Phase
  8. The Neural Network
  9. Input Encoding and Data Prediction
  10. Special case: triangles with hanging nodes
  11. The Approximation Spaces HNNj,E
  12. The Neural Network Architecture and the Training Phase
  13. An effective variant for the Training strategy
  14. Numerical Results
  15. Meshes and Training sets
  16. ...and 7 more sections

Key Result

Proposition 1

For all $E \in \mathcal{T}_{h\ifstrempty{}{}{_{}}}$ and for all $j=1,\dots,N^{\operatorname{dof}}\ifstrempty{E}{}{_{E}}$, it holds where $C_1$ and $C_2$ depends on $E$ and $\partial E$.

Figures (12)

  • Figure 1: Input Encoding. Left: Original elements. Center: Variability Reduction. Right: Input Reduction.
  • Figure 2: Examples of triangles with hanging nodes. The colors and labels are associated with the different configurations. The required rotations and reflections are already taken into account.
  • Figure 3: Left: Domain $\Omega_\Phi$ of the function $\Phi$. Right: Shape of the function $\Phi$.
  • Figure 4: Domain $\Omega_{\Phi,j,E}^{i}$ of the auxiliary function $\Phi^{i}_{j,E}$, for each $i=j-1,j,j+1$.
  • Figure 5: Left: First mesh of RDQM family. Center: First mesh of VM family. Right: First mesh of HTM family. The red dots denote the hanging nodes.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Proposition 2
  • proof