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

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.