The introduction of convolutional layers greatly advanced the performance of<br />
neural networks on image tasks due to innately capturing a way of encoding and<br />
learning translation-invariant operations, matching one of the underlying<br />
symmetries of the image domain. In comparison, there are a number of problems<br />
in which there are a number of different inputs which are all &#039;of the same<br />
type&#039; --- multiple particles, multiple agents, multiple stock prices, etc. The<br />
corresponding symmetry to this is permutation symmetry, in that the algorithm<br />
should not depend on the specific ordering of the input data. We discuss a<br />
permutation-invariant neural network layer in analogy to convolutional layers,<br />
and show the ability of this architecture to learn to predict the motion of a<br />
variable number of interacting hard discs in 2D. In the same way that<br />
convolutional layers can generalize to different image sizes, the permutation<br />
layer we describe generalizes to different numbers of objects.