My general observation is that these differential equations have difficult finding values that deal with a global max or min, in particular f'=0 mostly gives you the local max/min. Therefore any differential equation no matter how sophisticate and well thought thru, dealing with min/max problem e.g. fitting against a data minimizing the error, alone by itself as a differential equation is not adequate.

Just like newton raphson, you estimate a solution, that is not enough, you need another step to move the solution towards a final solution which ends up to be minimizing algorithm.

To solve this in general Differential Evolution has been devised by many researchers which then allows for a fast number of iterations to get to a GLOBAL MIN/MAX solution.