  A complete structure-preserving learning scheme for
single-input/single-output (SISO) linear port-Hamiltonian systems is proposed.
The construction is based on the solution, when possible, of the unique
identification problem for these systems, in ways that reveal fundamental
relationships between classical notions in control theory and crucial
properties in the machine learning context, like structure-preservation and
expressive power. In the canonical case, it is shown that the set of uniquely
identified systems can be explicitly characterized as a smooth manifold endowed
with global Euclidean coordinates, which allows concluding that the parameter
complexity necessary for the replication of the dynamics is only $O(n)$ and not
$O(n^2)$, as suggested by the standard parametrization of these systems.
Furthermore, it is shown that linear port-Hamiltonian systems can be learned
while remaining agnostic about the dimension of the underlying data-generating
system. Numerical experiments show that this methodology can be used to
efficiently estimate linear port-Hamiltonian systems out of input-output
realizations, making the contributions in this paper the first example of a
structure-preserving machine learning paradigm for linear port-Hamiltonian
systems based on explicit representations of this model category.
