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One type of deviation from another map goes under name of "optimal transport over linear/nonlinear dynamics'. Described in the continuous form of OT (Brenier-Benamou style), here the baseline dynamics are given either by a linear or nonlinear ode, and the job is to find the deviation to baseline that leads to desired transport in optimal way. This idea generalizes optimal control to optional transport of measures. Google scholar with these terms will help you.