Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations
This work addresses the challenge of data-driven dynamical system approximation for researchers in control theory and applied mathematics, offering an incremental improvement over existing methods like Extended Dynamic Mode Decomposition.
The paper tackles the problem of approximating unknown dynamical systems using Koopman-operator methods by proposing the Tunable Symmetric Subspace Decomposition algorithm to refine function dictionaries, balancing expressiveness and accuracy. Simulations on planar systems demonstrate its effectiveness in producing tunable Koopman approximations that capture relevant system information.
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.