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Two continuous extensions of the Neural Approximated Virtual Element Method

Stefano Berrone, Moreno Pintore, Gioana Teora

TL;DR

The paper introduces two globally continuous NAVEM variants, B-NAVEM and P-NAVEM, to enforce inter-element continuity for neural-approximated virtual element basis functions on polygonal meshes. B-NAVEM relies on a Physics-Informed Neural Network to enforce the local Laplace problem while boundary values are made exact via a boundary operator, whereas P-NAVEM emphasizes exact polynomial reproducibility through a boundary-based operator and tailored loss terms, potentially avoiding interior harmonicity. Both methods use a bubble- and transfinite-interpolation-based operator to ensure boundary conformity, enabling a consistent C^0-conforming discretization with neural interior corrections. Numerical experiments show that P-NAVEM generally offers the best accuracy-to-cost trade-off, with neural approaches achieving competitive errors and favorable performance in nonlinear problems, though training and per-element costs can be higher than standard VEM for coarse meshes. Overall, the work demonstrates viable pathways to continuity-preserving, neural-enabled VEM discretizations that cope with linear and nonlinear PDEs on general polygonal meshes, and it points to future extensions such as V-PINN variants and Zipped Finite Elements for higher-order spaces.

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.

Two continuous extensions of the Neural Approximated Virtual Element Method

TL;DR

The paper introduces two globally continuous NAVEM variants, B-NAVEM and P-NAVEM, to enforce inter-element continuity for neural-approximated virtual element basis functions on polygonal meshes. B-NAVEM relies on a Physics-Informed Neural Network to enforce the local Laplace problem while boundary values are made exact via a boundary operator, whereas P-NAVEM emphasizes exact polynomial reproducibility through a boundary-based operator and tailored loss terms, potentially avoiding interior harmonicity. Both methods use a bubble- and transfinite-interpolation-based operator to ensure boundary conformity, enabling a consistent C^0-conforming discretization with neural interior corrections. Numerical experiments show that P-NAVEM generally offers the best accuracy-to-cost trade-off, with neural approaches achieving competitive errors and favorable performance in nonlinear problems, though training and per-element costs can be higher than standard VEM for coarse meshes. Overall, the work demonstrates viable pathways to continuity-preserving, neural-enabled VEM discretizations that cope with linear and nonlinear PDEs on general polygonal meshes, and it points to future extensions such as V-PINN variants and Zipped Finite Elements for higher-order spaces.

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.
Paper Structure (14 sections, 61 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 61 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural networks to approximate basis functions
  3. The NAVEM formulation
  4. Enforcing continuity across adjacent elements
  5. A bubble function on a generic polygon
  6. The transfinite interpolation of $\varphi_{j,E}$
  7. The operator $\mathcal{B}_{j,E}$
  8. The B-NAVEM formulation
  9. The P-NAVEM formulation
  10. Numerical results
  11. Test 1: The neural networks and the training phase
  12. Test 2: A diffusion-advection-reaction problem
  13. Test 3: A nonlinear diffusion problem
  14. Conclusion

Figures (12)

  • Figure 1: Left: Distance function \ref{['eq:def_phi_line']} to the line lying on the segment of extrema $(-0.5, 0.0)$ and $(0.5, 0.0)$. Center: trimming function defined in \ref{['eq:trim_circle']} in related the the edge of extrema $(-0.5, 0.0)$ and $(0.5, 0.0)$. Right: the ADF function defined in \ref{['eq:def_phi_segment']} related the the edge of extrema $(-0.5, 0.0)$ and $(0.5, 0.0)$.
  • Figure 2: Different alternatives to define the bubble function $\psi_{E}^0$ for a convex polygon (left), a triangle with hanging nodes (center), and a concave polygon (right). Top row refers to definition \ref{['eq:bubble_convex']}, whereas the bottom row refers to definition \ref{['eq:def_phi_segment']}.
  • Figure 3: Tests 1, 2, and 3: Second mesh in the two considered families of meshes used in the test phase. Left: Voronoi. Right: Convex-Concave.
  • Figure 4: Points distribution generated by using the Algorithm \ref{['alg:points_on_triangle']} on general polygons.
  • Figure 5: Test 1: Behaviour of loss values as the number of training steps increases (left) and time increases (right) during the training phase.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5