Cross validation when training neural network?
I couldn't say what the authors refer to by


  best by cross-validation


but I'll mention a simple and general procedure that's out there:

You are correct that analyzing one estimate of the generalization performance using one training and one test set is quite simplistic. Cross-validation can help us understand how this performance varies across datasets, instead of wonder whether we got lucky/unlucky with our choice of train/test datasets. 

Split the whole dataset into k folds (or partitions), and train/test the model k times using different folds. When you're done, you can compute the mean performance and a variance that will be of utmost importance assessing confidence in the generalization performance estimate.