However, nonlinear functions may be approximated by using an appropriate nonlinear activation function 𝛼 𝑦(𝑥) that acts on the output, 𝑏 + 𝑤𝑋, of each neuron. Furthermore, the depth of the neural network allows for abstraction of the input. It has been shown that the early layers in a deep feed-forward neural network are primarily responsible for representing simple data patterns and gradients while the later layers are responsible for building abstraction and complexity. It may seem difficult to rationalize that large ensembles of perceptrons can be manipulated to produce a coherent output. However, the idea of complex behaviors originating from simple first principles calculations is not new. In fact, many of the same tools used to study the macroscopic behavior of elementary operators from theoretical physics have been applied to study DL models corresponding to the number of space groups, extinction groups, and crystal systems respectively.