NN multidim regression with matrix as output
This is the answer to a question regarding unknown labels.
You have to know labels before using any supervised algorithm. Otherwise, there is no way you can train a model. You need to think of solving this problem by employing one of the unsupervised techniques, such as k-means algorithm, Gaussian Mixture Models, or Classification And Regression Trees, etc.
For example, one of the suggestions I would give is to try k-means with a fixed number of k, which is 1000 in your case, run the algorithm a couple of times and see if centroids are any close to elements in your output. Then, you could classify your output based on how close they are to one of the nearest centroids. Then all elements of input belonging to individual centroids would be classified as such.

EDIT.
After reconsidering your example, I think that perhaps k-NN would be much more helpful to your problem. In k-NN, outputs are considered as neighbours. Each point from the input is assigned to the closest neighbour. In the end, you already have the output but you don't know how to map all elements in both the input and the output. I have just realised your problem is "the mapping". It is a good chance k-NN would solve that meaning it would create labels for all elements in the input that correspond to elements in the output. Once it is done, the Neural Network model can be trained.
I want to build a simple NN for regression purposes; the dimension of my input data reads (100000,3): meaning I have 1mio particles and their corresponding x,y,z coordinates. Out of these particles I want to predict centers which the particles correspond to where the data of the centers read (1000,3).
my question is: since the input array should have the same number of samples as target arrays how can I solve this problem?
actually my mapping is from (100000,3) -> (1000,3) because on average about 100 particles belong to one center.
to train the model i will use many of those datasets with the right centers as output; after that i want to predict out of one new set of particle coordinates the corresponding centers.