Classifying with certainty I'm trying to classify a binary sample with Keras and I would like to classify as many correctly as possible, while ignore the ones where the model is not sure.

The fully connected Nerual network currencly achieves around 65% but I would like to get a higher result of correctly classified ones, while ignoring the ones where the model is uncertain.

Is there a way to tell Keras to simply ignore the ones where the model is uncertain and achieve a higher accuracy that way? Or is there a network design that could achieve this, for example feeding the result of the network striaght into a second part of it which then decides whether the prediction is likely accurate or not?

One way I was thinking of achieving this is by building a second neural network on top of it that decides based on the result of the first network and all the input data of it, whether the classification will be correct or not. Would that work, and if yes, is there no more elegant way of achieving this in one go, such as directly having the results feeding into a second part of the network that then decides if the prediction is likley accurate or not?
Softmax output in neural networks can be misleading - often the confidence provided is higher than is intuitive.  See e.g. here:

A Baseline for Detecting Misclassified and
Out-of-Distribution Examples in Neural Networks
https://pdfs.semanticscholar.org/c26e/1beaeaa55acae7336882de5df48716afb8bb.pdf

which suggests that in practice, softmax is not helpfully interpretable as a probability but should instead be used for ranking among class options.

If you want to have an accurate probability estimate, you might consider using a Bayesian approach in which you explicitly model your estimate of each of the input variances, the output variance, etc.

Failing that, having a second phase neural network that takes the input and predict correct or incorrect classification by the first network is an interesting idea - where incorrect classification is a proxy for 'low confidence' classification.  If you try it I'd be curious to know how it works.

Edit: As @Emre said the input to the softmax would be more informative than the softmax itself because it's pre-scaled (i.e. not forced to sum to 1). So it should reflect confidence better, with values further away from 0 indicating higher confidence.