So first some context. I am currently trying to figure out how to estimate the solution of a PDE by using an observer, and I think I can work it by using a Kalman filter to get some measurements, and then use that to utilize an ML/kernel based observer to predict the solution (observer described here with the code, and this github repository has all the functions used). Here, what they do is use a data file with the variables orig_func_data, orig_func_obs, and orig_func_plot_vals to run the code. Hopefully I'm thinking of this correctly, but I'm assuming that, in the case of the heat equation $$u_t = \alpha u_{xx},$$ the "data" variable is the $x$ values of the function, the "observed" would be from using something like a Kalman filter (which is an observer, right?), and then the "plot_vals" one would be the $u$ values, while this whole process moves forward in time.
I used finite differences to get solution values for the heat equation for a given $\alpha$ and initial conditions (so I got my plot vals variable), but I'm not sure how to get observed values, so I want to try to use a Kalman filter, for which I think I should put it in state-space representation. But, I think I'm having trouble figuring out if this is correct? I haven't ever studied control theory explicitly, and I'm doing this for some other work, so I honestly have no idea if what I'm doing is reasonable or not.
What I did is use separation of variables on the heat equation to get $$T(t)' = \lambda T(t)\\X''(x) = \frac{\lambda X(x)}{\alpha}.$$ But, I'm not sure if I can get this into state space form, since they're clearly not differentiable by the same variable. Am I thinking of this the correct way, or should I be trying a completely different approach?