I have read this version of the definition of functional derivative https://en.wikipedia.org/wiki/Functional_derivative#Functional_derivative
In the above definition the functional derivative takes values in $\mathbb{R}$, and the arguments of the functional are perturbed an amount $\epsilon$ like so : $\rho+\epsilon \phi$.
Now, in what im reading ( the appendix Lemma 1 ) https://arxiv.org/pdf/2003.03803.pdf the functional derivative takes values in $\mathbb{R}^d$ and the perturbation is done via a flow problem. I'm confused by their working :
Can anyone help point me towards the definition of the $\frac{\delta\mathcal{E}}{\delta X}$ they are using, and how they apply this chain rule? Mainly how do I reconcile perturbations of the form $X+sY$ (as seen in the definition of functional derivative) with those of the form $\partial_sX^s=v \circ X^s$ ?