I have been working on using automatic differentiation to compute the sensitivities of systems of ordinary differential equations of the form
\begin{equation} \frac{d\mathbf{u}(t)}{dt} = F(\mathbf{u}(t),t,\alpha),\qquad \mathbf{u}(0) = \mathbf{u}^0. \end{equation} where $\alpha\in\mathbb{R}^M$ is a vetor of real valued parameters and $\mathbf{u}(t)\in\mathbb{R}^N$.
To showcase the advantages of the algorithm, I was looking for a real scenario where the vector valued function $F$ depends on a number of parameters $M$ much bigger than the number of equations in the system $N$. By real, I mean some equation with applicability in any scientific field.
In the article "Neural Ordinary Differential Equations" by Ricky T. Q. Chen, they consider $F$ to be a neural network, where indeed $M \gg N$, but I was looking for an alternative.
What I have considered so far are PDEs discretized in space using finite differences, for example, the 1D heat equation with spatial dependent thermal diffusivity $\alpha(x)$. However, if I consider the set of parameters $\alpha_i = \alpha(x_i)$, where $x_i$ are the grid points, that would result in $M=N$. I could artificially consider more parameters, like $\alpha_{i+\frac{1}{2}} = \alpha(x_{i+\frac{1}{2}})$ and force these on the function $F$, but still I would get $M=2N$, and I don't think it's usual for numerical methods to consider a finer grid for the parameters, rather the other way around (thinking of FEM for example, where $\alpha(x)$ could possibly be represented by its mean value in the element).
Another thing I have considered is if $F$ depends on some function which needs to be numerically evaluated, and thus would be approximated by a finite set of $M$ parameters ( for example, coefficientes for a given set of basis functions). Can you think of any real example like this where $M \gg N$?
What I really really want is something like $M \approx N^2$ or at least $M \approx kN$ with $k \gg 1$.