Koopman Regularization and Independent Koopman Eigenfunction Approximation

Abstract

This paper provides two algorithms to extract the governing equations of the dynamical system from samples, \acf{KR} and \acf{IKEA}. These algorithms extract a functionally independent set of Koopman Eigenfunctions. This kind of set is the most informative since it reveals the geometric structure of the dynamical system. Consequently, despite its finite cardinality, the governing equations are immaculately restored. Thus, the principle of parsimony is implemented. Samples can be taken either from a vector field itself or from orbits of the system. The algorithm \acl{KR} extracts the governing law from samples of the vector field, and the algorithm \acl{IKEA} extracts it from samples of trajectories of the system. Both of these algorithms are constrained optimization-based methods for learning the governing equations from sparse and corrupted samples of the vector field or trajectories. The objective functional, based on the Koopman Partial Differential Equation, approximates the Koopman Eigenfunctions relying on the samples, and the feasible functional forces this set to be functionally independent. The condition for functional independence is thoroughly discussed and formulated here, based on the Gershgorin Circle Theorem. For both algorithms, the optimization process is induced by the penalty method with backtracking that yields promising results in denoising, generalization, and dimensionality reduction, and shows better results than \acs{DMD}, Extended \acs{DMD}, \acs{PINN}, and \acs{SINDy} in dynamical system restoration.