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In Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more Sanz Serna writes:

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?

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  • $\begingroup$ 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. $\endgroup$ – GoHokies Jun 26 at 19:13
  • $\begingroup$ @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. $\endgroup$ – homocomputeris Jun 27 at 21:15
  • $\begingroup$ 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). $\endgroup$ – GoHokies Jun 28 at 5:26

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