Problem
At the moment, I have a problem with seven variables:
$S, A_1, A_2, R_1, R_2, P_0, P_1 $ and $P_2$.
Each of these variables draws a smooth line through time. My question is, is there any proof out there that there exists a differential equation model that will fit all of these functions perfectly.
More specifically, is it possible to define a differential equation model of form:
$$ \frac{dS}{dt} = f(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dA_1}{dt} = g(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dA_2}{dt} = h(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dR_1}{dt} = i(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dR_2}{dt} = j(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dP_0}{dt} = k(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dP_1}{dt} = l(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
$$ \frac{dP_2}{dt} = m(S, A_1, A_2, R_1, R_2, P_0, P_1, P_2) $$
That will model each of these curves, with no error, regardless of their shape, given they are smooth.
Context
I am trying to derive a differential equation model for some data of this form using machine learning techniques. It is possible to specify a good approximation of a differential equation that will generate each of the curves individually (so for example, we can find a good approximation of dS/dt that will reliably generate a curve similar to the underlying S curve when given the true values of the other variables). However, these equations are very complex as they are essentially trying to build curves out of arbitrary data, as there is not necessarily any underlying relationship between the seven variables.
When you build the model from initial conditions, using these differential equations and without access to the underlying data, we encounter problems as they are so complex they are very sensitive. As the functions generated through the machine learning algorithm are only good approximations, small errors begin to propagate through the entire system. This then grows due to the complexity of the functions and makes the model highly inaccurate.
What I am hoping for is some proof that shows that, actually, it is not possible to attempt this and build a stable system.
Thanks in advance!