PFEM-GP-dPHS : a finite element framework for combining Gaussian processes and infinite-dimensional port-Hamiltonian systems

In order to learn distributed port-Hamiltonian systems (dPHS) using Gaussian processes (GPs), the partitioned finite element method (PFEM) is combined with the Gp-dPHS method. By following a late lumping approach, the discretization of the functional hyperparameters of the GP prior over the Hamiltonian functional is chosen independently from the discretization of the dPHS, thus reducing the numerical complexity of our method. We next model the mean of the GP prior of the Hamiltonian as a quadratic form, enabling the GP kernel to focus on the nonlinear part of a given dPHS. We illustrate our method on a nonlinear one dimensional wave equation with unknown physical parameters (tension and linear mass).