Convergence of data-driven Koopman-operator logarithms to generators in general nonlinear systems

Determine how the data-driven approximation of the logarithm of the Koopman operator log(K_t) converges to the infinitesimal generator L of the Koopman semigroup for continuous-time nonlinear dynamical systems that do not satisfy the bounded-generator and spectral-sector constraints required for single-valued operator logarithms; specifically, establish convergence guarantees for L = (1/t) log(K_t) when L may be unbounded and the spectrum of K_t is not confined to a sector in which the logarithm is single-valued.

Background

In data-driven identification of continuous-time nonlinear systems, one indirect approach estimates the infinitesimal generator L by computing the operator logarithm of a learned Koopman operator K_t via the relation L = (1/t) log(K_t). This Koopman logarithm method avoids estimating time derivatives from high-frequency observations but typically requires that L be bounded and that the spectrum of K_t lies in a sector where the operator logarithm is single-valued.

The authors note that for general systems outside these restrictive conditions, the convergence of the data-driven approximation of log(K_t) to the true generator L is not established. This motivates their development of a resolvent-type, logarithm-free learning scheme, while leaving open the question of convergence for the logarithm-based approach in the broader, unbounded-generator setting.