Covariance matrix size, multitask GP

[sanaamouzahir]

Hi! I am currently working on applying Gaussian process in dynamical systems modeling, and my team and I have been having a confusion regarding the shape of the covariance matrix.
Consider Nr realizations of an experiment (Say same experiment with different initial conditions each time), each experiment is of time T,with Nt time steps, for Nx grid points. My thoughts are that the shape of the covariance matrix is (NrNtNx)(NrNt*Nx), as the covariance matrix characterises the covariance between the outputs (In my case the velocity evaluated at the grid points at given times). Is that correct?
Or should the covariance matrix just scale with the number of time steps and number of grid points?

Thank you!