Koopman Regularization
Ido Cohen
TL;DR
Koopman Regularization addresses learning governing vector fields from sparse and corrupted samples by exploiting the geometry of Koopman eigenfunctions. It reformulates the KPDE as an optimization objective while enforcing functional independence as a geometric constraint through a barrier-method approach, enabling denoising, generalization, and dimensionality reduction. A minimal set of $N$ functionally independent eigenfunctions (or UVMs) yields a parsimonious yet exact representation of the dynamics. Experiments on linear/nonlinear 2D systems and the Lorenz butterfly demonstrate effective denoising, accurate generalization from sparse data, and compact dimensionality reduction, with measurable improvements in SNR and MSE.
Abstract
\emph{Koopman Regularization} is a constrained optimization-based method to learn the governing equations from sparse and corrupted samples of the vector field. \emph{Koopman Regularization} extracts a functionally independent set of Koopman eigenfunctions from the samples. This set implements the principle of parsimony, since, even though its cardinality is finite, it restores the dynamics precisely. \emph{Koopman Regularization} formulates the Koopman Partial Differential Equation as the objective function and the condition of functional independence as the feasible region. Then, this work suggests a barrier method-based algorithm to solve this constrained optimization problem that yields promising results in denoising, generalization, and dimensionality reduction.