# Reverse automatic differentiation and integration

It is well known that the reverse mode of differentiation implies an integration of the adjoint equations.

Any source that shows the fact explicitly? If I want to differentiate $$z = f(x, y)$$ w.r.t. $$(x,y)$$, what would be the adjoint equation that should be integrated?

• this is all explained on page 2 of that paper, see the paragraph that starts with Section 3, the core of the paper (...). as for proofs re. adjoint consistency of RK discretizations, Andrea Walther's paper is a good place to start from. Jun 26 '19 at 19:13
• @GoHokies They show how to get gradient w.r.t. the initial conditions, I don't get what the adjoint equation should be if there is no DE constraints and no cost function to minimize. Jun 27 '19 at 21:15
• if you're looking at how to differentiate more general expressions $z = f(x,y)$ wrt $(x,y)$ then have a gander at this question (and the paper I refer to therein). Jun 28 '19 at 5:26