Which optimization method can be used to do the following?

Question

I've the following system of equations for studying information flow in the below graph,

$$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise effects} \hspace{1cm} (1)$$

Here, M is the incidence matrix of the graph

$\phi$ is a vector with variables [ A B C D E F].

I've solved the above ode's to obtain the time series data of variables A,B,C,D,E,F.

Using the time-series data obtained from the above step, I'd like to do determine $\tilde{D}$ for the following system

$$ \frac{d \phi}{dt} = -M^T\tilde{D}M\phi \hspace{1cm} (2)$$

Note: The entries in the diagonal elements of $\tilde{D}$ are the edge weights.

I'd like to know if there is a way to solve this as an optimization problem using time-series data as input.

Any suggestions on how to proceed will be really helpful

EDIT: Equation (1) (with noise effects) is solved using prior values of the diagonal matrix, D and the time series profiles of variables in each node are obtained.

What do I want to achieve?

I want to determine a modified D i.e $\tilde{D}$ that can generate the same time series profile that was generated while solving equation (1).

EDIT: Based on the below discussion

The cost function that I want to minimize is

$$\int_{0}^{t_{f}} || \phi(t) - \exp{(-M^{T} \tilde{D} M t)} \phi(0) ||^{2} dt$$

to reduce the difference between the solutions of (1) and (2).

Any suggestions on the optimization function and toolbox that can be used to solve the above will be really helpful

EDIT3: I looked at some literature and I think this problem has to be solved using fmincon and it might not be possible to use convex optimization.

I'm still looking for suggestions on how to formulate the constraints for solving the cost function illustrated below.

Answer 1

Despite my comment, I think you can find $\tilde{D}$ that contains the noise term as well. You have this equation:

$$-M^{T} \tilde{D} M \phi(t) = -M^{T} D M \phi(t) + W(t)$$

Where $W(t)$ is the noise term vector. So:

$$-M^{T} (\tilde{D}-D) M \phi(t) = W(t)$$

Take $\mathcal{D} = \tilde{D} - D$.

Let's expand this equation:

$$(M \phi(t))_{i} = \sum_{j=1}^{6} M_{ij} \phi_{j}$$

$$(\mathcal{D} M \phi(t))_{i} = \sum_{j=1}^{6} \sum_{k=1}^{6} \mathcal{D}_{ij} M_{ik} \phi_{k}$$

But $\mathcal{D}$ is diagonal, which means:

$$\mathcal{D}_{ij} = \mathfrak{D}_{ii}\delta_{ij}$$

Where $\delta_{ij}$ is the Kronecker delta. So:

$$(\mathcal{D} M \phi(t))_{i} = \sum_{j=1}^{6} \sum_{k=1}^{6} \mathcal{D}_{ij} M_{ik} \phi_{k} = \sum_{j=1}^{6} \sum_{k=1}^{6} \mathfrak{D}_{ii} \delta_{ij} M_{ik} \phi_{k} = \mathfrak{D}_{ii} \sum_{k=1}^{6} M_{ik}\phi_{k}$$

Finally:

$$(M^{T} \mathcal{D} M \phi(t))_{i} = \sum_{j=1}^{6} M_{ji} \mathfrak{D}_{jj} \sum_{k=1}^{6} M_{jk}\phi_{k} = \sum_{j=1}^{6} \sum_{k=1}^{6} M_{ji}M_{jk} \phi_{k} \mathfrak{D}_{jj}$$

or:

$$-(M^{T} \mathcal{D} M \phi(t))_{i} = (W(t))_{i}$$

so:

$$\sum_{j=1}^{6} \sum_{k=1}^{6} M_{ji}M_{jk} \phi_{k} \mathfrak{D}_{jj} = -W_{i}$$

or in matrix form by defining the vector $\mathsf{x} = (\mathfrak{D}_{11},\mathfrak{D}_{22},\mathfrak{D}_{33},\mathfrak{D}_{44},\mathfrak{D}_{55},\mathfrak{D}_{66})^{T}$ and matrix $\mathsf{A}_{ij} = M_{ji}\sum_{k=1}^{6} M_{jk} \phi_{k}$:

$$\mathsf{A} \mathsf{x} = -W$$

Note that $\mathsf{A}$ and $W$ depends on time, so for each time you need to solve this equation to extract $\mathsf{x}$. In fact due to the fact that $\tilde{D}$ suppose to contain the noise term as well, it must depend on time.

So, you can easily find $\mathsf{x}$ by solving this linear equation and it would be a unique solution. The only comment here is that: this approach is somewhat looks like a reverse engineering and for a calculated $\phi$ and having that particular $W(t)$ noise term, you embed it into the diagonal matrix $D$.

Update: Another approach to not using an exact method is compromising the accuracy of $\phi(t)$ to find a constant $\tilde{D}$. The exact equation is:

$$\phi^{'}(t) = -M^{T} D M \phi(t) + W(t)$$

and the approximate equation is:

$$\hat{\phi}^{'}(t) = -M^{T} \tilde{D} M \hat{\phi}(t)$$

Now we want to minimize this:

$$\mathsf{K} = \int_{0}^{t_{f}} ||\phi(t) - \hat{\phi}(t)||^{2} dt$$

Analytically for $\hat{\phi}(t)$:

$$\hat{\phi}(t) = \exp{(-M^{T} \tilde{D} M t)} \hat{\phi}(0)$$

So finally your optimization problem is:

$$\tilde{D}, \hat{\phi}(0) = \text{argmin} \ \mathsf{K}(\tilde{D},\hat{\phi}(0)) = \text{argmin} \ \int_{0}^{t_{f}} ||\phi(t) - \exp{(-M^{T} \tilde{D} M t)} \hat{\phi}(0)||^{2} dt$$

This problem could be solved by CVXPY tool I believe.