some comprehension questions for iLQR method in this work

[Guanmupp]

Very amazing job, but i'm sorry to raise some comprehension questions that i'm stucked with. Given that ilqr need to resolve the Qu,Quu,Qxu by fu,fx. In this work ,the input u has been fisrtly fed into a NN then influence the x，then how can the automatic differentiation in jax describe the connection between status x and input u.
Hope for instruction.
Best regrads.