  Modeling phenomena from experimental data, always begin with a \emph{choice
of hypothesis} on the observed dynamics such as \emph{determinism},
\emph{randomness}, \emph{derivability} etc. Depending on these choices,
different behaviors can be observed. The natural question associated to the
modeling problem is the following : \emph{"With a finite set of data concerning
a phenomenon, can we recover its underlying nature ?} From this problem, we
introduce in this paper the definition of \emph{multi-scale functions},
\emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale
calculus} (see \cite{bohn}). These definitions will be illustrated on the
\emph{multi-scale Okamoto's functions}. The introduced formalism explains why
there exists different continuous models associated to an equation with
different \emph{scale regimes} whereas the equation is \emph{scale invariant}.
A typical example of such an equation, is the \emph{Euler-Lagrange equation}
and particularly the \emph{Newton's equation} which will be discussed. Notably,
we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton's
equation} and also the \emph{non-linear Schr\"odinger equation} via the
\emph{scale Newton's equation}. Under special assumptions, we recover the
classical \emph{diffusion} equation and the \emph{Schr\"odinger equation}.
