Isn't Runge-Kutta a method for numeric integration of known curves? I think essentially you're suggesting to regress a high-order Runge-Kutta and then minimize the differential area. That's an interesting approach, but I suspect pretty computationally slow compared to directly fitting splines or something like that. I suspect the OP would be better off doing direct interpolation instead, especially because Runge-Kutta won't allow for extrapolation beyond predicted values.

One other note: remember that if you try to estimate a high-order equation without a lot of input data, you can run into over-fitting problems. Most numerical techniques get much more accurate (and slower) with lots of data