I don't know if my question is legitimate, or if it is meaningless :
Newton's method consist in solving the non linear equation : $F(\mathbf{x})=0$ in proceeding to the iteration : $\mathbf{x}_{n+1}=\mathbf{x}_n-\mathbf{J}^{-1}(\mathbf{x_n})F(\mathbf{x_n})$.
It reminds of a time dependent process where $n$ is the time. $\mathbf{x}(t+\Delta t)=\mathbf{x}(t)-\mathbf{J}^{-1}(\mathbf{x}(t))F(\mathbf{x}(t))$.
Then I'm wondering if there is a continuous approach, meaning : $\frac{\Delta\mathbf{x}(t)}{\Delta t}=\frac{\mathbf{J}^{-1}(\mathbf{x}(t))F(\mathbf{x}(t))}{\Delta t} $.
The goal would be to solve a continuous equation, which could be easier in some cases that implementing a discrete process.
But I don't know how to continue.
Does it make sense ?