I have a Deep Neural Network that takes $n$ inputs $X = [X_1, \ldots, X_n]^T$ and gives $n$ ouputs $Y = [Y_1, \ldots, Y_n]^T$. Normally, I can just do a standard deep neural network with a few fully connected hidden layers. However, I want to build into the network the fact that there should be permutational symmetry. That is to say, if $\pi(\cdot)$ is a particular permutation, then $\pi(X)$ should output $\pi(Y)$. So each input should be treated symmetrically in some sense.
A simple example would be I get $n$ IID sensor readings, and I want to build an auto-encoder for these sensor readings.
Is there some way of building this symmetry into the neural network, perhaps through some parameter sharing or a special architecture?