๐ค AI Summary
This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.
๐ Abstract
This paper introduces shadowing theory based backward stability assessment of data driven computational analysis of discrete dynamical systems in the framework of the Koopman composition operator and the Extended Dynamic Mode Decomposition. Data driven spectral analysis of the dynamics using the approximate (computed) eigenpairs of the Koopman operator is cast in terms of the backward error analysis. The individual residuals of the computed eigenpairs are aggregated in a backward perturbation of the Koopman operator, so that the computed eigenpairs are exact for a perturbed operator that is not a composition operator. Then, the impact of the perturbation in the operator is carried over to the map and the initial condition of the original system, showing that the computed approximations correspond exactly to a pseudo--trajectory of the system. In the final step, the pseudo--trajectory is shadowed by an exact trajectory of the system. This interpretation of errors is in particular suitable in data driven scenarios where the data is contaminated by noise. New insights into the numerical shadowing are provided.