Essentially, you transform your function into a “circuit” or just a graph with edge labels according to the relationship between parts of the expression. The circuit has the nice property that there is an algorithm you can run on it, with very simple rules, which gets you the derivative of the function used to create that circuit.
So taking the derivative becomes:
1. Transform function F into circuit C. 2. Run compute_gradiant(c) to get the gradient of F.
Lots of useful examples here: https://cs231n.github.io/optimization-2/
Of course you can, what do you think shift operators and recurrence relations are? https://en.wikipedia.org/wiki/Finite_difference?#Calculus_of...
We do calculus to predict behavior in fields.
We observe metrics and conservational symmetry (or not) over paths in fields.
Nonlinearity is approximated with backpropagation.
What are field operators (graph operators)?