Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of 10000 to 20000. Also all entries of the vectors are real numbers between $0$ and $1$, if that helps.

We now consider the system of linear ODEs \begin{equation} \frac{d}{dt} x = A \cdot x + b, \end{equation} with respective initial vectors $x^{(1)},\ldots,x^{(r)}$, where $A$ is a real valued $n \times n$ matrix and $b$ is a real $n$-dimensional vector. However, we do not know the coefficients of $A$ and $b$ yet. (If beneficial, you may assume $b=0$.)

Consider now the function $G: \mathbb{R}^n \rightarrow \mathbb{R}$ given by \begin{equation} G(x) = c_0 + \sum_{i=1}^n c_i \cdot x_i. \end{equation} Here most $c_i$, i.e. all apart from 6 to 12 of these $c_i$, are 0, so just a fraction of the entries of the vectors matters. We do know the exact weights of the $c_i$.

We fix three time points: Time point zero plus two further time points $(t_0,t_1,t_2)$.

Now let $x^{(j)}(t_k)$ denote the vector of the curve induced by $x^{(j)}$ and the system of linear ODEs at time point $t_k$. Then, in our scencario we do have for $i = 1,\ldots,r$ and $k=0,\ldots,2$, the values $G(x^{(j)}(t_k))$ at hand, which is, of course, a much reduced information compared to the complete knowledge of $x^{(j)}(t_k)$ and also just allows limited insight, since most entries do not matter in $G$.

My question is now: Can we fit the entries $A$ (and $b$) by knowing all $x^{(j)}$ initial vectors, the function $G$ and the values $G(x^{(j)}(t_k))$ for $i = 1,\ldots,r$ and $k=0,\ldots,2$ ?

On the one hand, since $G$ uses just a small fraction of entries, this seems to be very difficult (particularly for predicting the dynamics of those entries that do not matter in $G$), but on the other hand we have around 10000 to 20000 measured values for each time point and just 10000 entries in the matrix $A$ to fit. Therefore, there might be a chance to fit it.