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

TL;DR

This paper introduces a neural network-enhanced finite element method for computing Ginzburg-Landau energy minimizers, capable of handling various parameters and improving efficiency.

Contribution

It presents a novel unsupervised deep Ritz-type neural network approach that can serve as a standalone solver or as an initializer for classical methods.

Findings

Neural network approach effectively computes energy minimizers.

Method handles a range of parameter values for $$-values.

Numerical examples demonstrate the strategy's potential.

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 $\kappa$ as a variable input parameter to obtain possible minimizers for a large range of $\kappa$-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.