  This paper tackles the data-driven approximation of unknown dynamical systems
using Koopman-operator methods. Given a dictionary of functions, these methods
approximate the projection of the action of the operator on the
finite-dimensional subspace spanned by the dictionary. We propose the Tunable
Symmetric Subspace Decomposition algorithm to refine the dictionary, balancing
its expressiveness and accuracy. Expressiveness corresponds to the ability of
the dictionary to describe the evolution of as many observables as possible and
accuracy corresponds to the ability to correctly predict their evolution. Based
on the observation that Koopman-invariant subspaces give rise to exact
predictions, we reason that prediction accuracy is a function of the degree of
invariance of the subspace generated by the dictionary and provide a
data-driven measure to measure invariance proximity. The proposed algorithm
iteratively prunes the initial functional space to identify a refined
dictionary of functions that satisfies the desired level of accuracy while
retaining as much of the original expressiveness as possible. We provide a full
characterization of the algorithm properties and show that it generalizes both
Extended Dynamic Mode Decomposition and Symmetric Subspace Decomposition.
Simulations on planar systems show the effectiveness of the proposed methods in
producing Koopman approximations of tunable accuracy that capture relevant
information about the dynamical system.
