Cluster identification with NN
If you want to treat clustering as a classification problem, then you can try to train the network to predict whether two points belong to the same clusters or to different clusters.
This does not ultimately solve your problems, though - to cluster the data, this labeling needs to be transitive (which it likely will not be) and you have to label n² pairs, which is expensive.
Furthermore, because your clustering is density-based, your network may need to know about further data points to judge which ones should be connected...
I have a dataframe containing the coordinates of millions of particles which I want to use to train a Neural network. These particles build individual clusters which are already identified and labeled; meaning that every particle is already assigned to its correct cluster (this assignment is done by a density estimation but for my purpose not that relevant).
the challenge is now to build a network which does this clustering after learning from the huge data. there are also a few more features in the dataframe like clustersize, amount of particles in a cluster etc. 
since this is not a classification problem but more a identification of clusters-challenge what kind of neural network should i use? I have also problems to build this network: for example a CNN which classifies wheather there is a dog or cat in the picture, the output is obviously binary. so also the last layer just consists of two outputs which represent the probability for being 1 or 0. But how can I implement the last layer when I want to identify clusters? 
during my research I heard about self organizing maps. would these networks do the job?
thank you