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GLENN: Neural network-enhanced computation of Ginzburg-Landau energy minimizers

Michael Crocoll, Christian Döding, Benjamin Dörich, Roland Maier

Abstract

In this work, we propose a neural network-enhanced finite element strategy to compute the minimizer of the Ginzburg--Landau energy based on an unsupervised deep Ritz-type strategy. We treat the parameter $κ$ as a variable input parameter to obtain possible minimizers for a large range of $κ$-values. This allows for two possible strategies: 1) The neural network may be extensively trained to work as a stand-alone solver. 2) Neural network results are used as starting values for a subsequent classical iterative minimization procedure. The latter strategy particularly circumvents the missing reliability of the neural network-based approach. Numerical examples are presented that show the potential of the proposed strategy.

GLENN: Neural network-enhanced computation of Ginzburg-Landau energy minimizers

Abstract

In this work, we propose a neural network-enhanced finite element strategy to compute the minimizer of the Ginzburg--Landau energy based on an unsupervised deep Ritz-type strategy. We treat the parameter as a variable input parameter to obtain possible minimizers for a large range of -values. This allows for two possible strategies: 1) The neural network may be extensively trained to work as a stand-alone solver. 2) Neural network results are used as starting values for a subsequent classical iterative minimization procedure. The latter strategy particularly circumvents the missing reliability of the neural network-based approach. Numerical examples are presented that show the potential of the proposed strategy.
Paper Structure (22 sections, 1 theorem, 46 equations, 5 figures, 6 tables, 1 algorithm)

This paper contains 22 sections, 1 theorem, 46 equations, 5 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Ginzburg-Landau problem and discrete minimizers
  3. Preliminaries
  4. A reduced Ginzburg--Landau model with given $\mathbf{A}$
  5. Numerical computation of Ginzburg--Landau minimizers
  6. Neural networks
  7. Machine learning basics
  8. MLPs, training, optimizers, and more
  9. Batches and epochs
  10. Unsupervised learning
  11. Machine learning for the Ginzburg--Landau problem
  12. Loss functions
  13. Neural network architecture
  14. Hardware
  15. Learning rate
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $\Omega \subset \mathbb{R}^d$, $d = 2,3$ be a bounded, simply connected Lipschitz domain and $\mathbf{h}_{\mathrm{ext}} \in L^2(\Omega,\mathbb{R}^{\ell_d})$. Then, the following statements hold.

Figures (5)

  • Figure 1: Block schematic of a SwiGLU block (left) and a DAGLU block (right).
  • Figure 2: Densities $|u|^2$ of computed minimizers for $\kappa = 10 ,25, 50, 75, 100$ (left to right), corresponding to Table \ref{['tab:energy_only_ord']}. Dark indicates values close to 1; light indicates values close to 0. First row: reference minimizer computed with the classical FE iterative solver and best heuristic initial values $\varphi_j$ (best). Second and third row: pure NN models GLENN-R1 (NN) and GLENN-R2 (NN). Fourth and fifth row: hybrid approach GLENN-R1 (hyb.) and GLENN-R2 (hyb.) using initial values from the NN.
  • Figure 3: Densities $|u|^2$ of computed minimizers for $\kappa = 10 ,25, 50, 75, 100$ (left to right), corresponding to Table \ref{['tab:energy_full']}. Dark indicates values close to 1; light indicates values close to 0. First row: reference minimizer computed with the classical FE iterative solver and best heuristic initial values $\varphi_j$ (best). Second and third row: pure NN models GLENN-F1 (NN) and GLENN-F2 (NN). Fourth and fifth row: hybrid approach GLENN-F1 (hyb.) and GLENN-F2 (hyb.) using initial values from the NN.
  • Figure 4: Densities $|u|^2$ of computed minimizers for $\kappa = 10 ,25, 50, 75, 100$ (left to right), corresponding to Table \ref{['tab:unit_square']}. Dark indicates values close to 1; light indicates values close to 0. First row: reference minimizer computed with the classical FE iterative solver and best heuristic initial values $\varphi_j$ (best). Second row: best hybrid approach using the fast-trained GLENN-U* (hyb.), $\texttt{*} \in \{ 1, 2, 3, 4, 5\}$.
  • Figure 5: Densities $|u|^2$ of computed minimizers for $\kappa = 10 ,25, 50, 75, 100$ (left to right), corresponding to Table \ref{['tab:Lshape']}. Dark indicates values close to 1; light indicates values close to 0. First row: reference minimizer computed with the classical FE iterative solver and best heuristic initial values $\varphi_j$ (best). Second row: best hybrid approach using the fast-trained GLENN-L* (hyb.), $\texttt{*} \in \{ 1, 2, 3, 4, 5\}$.

Theorems & Definitions (1)

  • Proposition 2.1