Resolvent-Type Data-Driven Learning of Generators for Unknown Continuous-Time Dynamical Systems

TL;DR

This work introduces a resolvent-type, data-driven framework for learning the infinitesimal generator $\mathcal{L}$ of the Koopman semigroup from discrete observations of unknown continuous-time systems. By leveraging Yosida approximations and finite-dimensional resolvent representations, the approach avoids operator logarithms and derivative estimation, enabling accurate generator learning under lower sampling rates. A practical data-driven algorithm is developed using a dictionary $Z_N$, distributed quadrature, and a two-resolvent strategy to estimate $\mathcal{L}_\lambda$ with convergence guarantees. The method is validated on multiple benchmarks (scaled Lorenz, polynomial, rational, biochemical, and power-system models) and demonstrates superior accuracy and robustness to derivative-based and log-based methods, including reliable region-of-attraction predictions via Zubov-type PDE constructions. Overall, RTM broadens Koopman-based learning to unknown, possibly unbounded generators and offers practical tools for system identification and stability analysis in complex dynamical systems.

Abstract

A semigroup characterization, or equivalently, a characterization by the generator, is a classical technique used to describe continuous-time nonlinear dynamical systems. In the realm of data-driven learning for an unknown nonlinear system, one must estimate the generator of the semigroup of the system's transfer operators (also known as the semigroup of Koopman operators) based on discrete-time observations and verify convergence to the true generator in an appropriate sense. As the generator encodes essential instantaneous transitional information of the system, challenges arise for some existing methods that rely on accurately estimating the time derivatives of the state with constraints on the observation rate. Recent literature develops a technique that avoids the use of time derivatives by employing the logarithm of a Koopman operator. However, the validity of this method has been demonstrated only within a restrictive function space and requires knowledge of the operator's spectral properties. In this paper, we propose a resolvent-type method for learning the system generator to relax the requirements on the observation frequency and overcome the constraints of taking operator logarithms. We also provide numerical examples to demonstrate its effectiveness in applications of system identification and constructing Lyapunov functions.

Figures (5)

• Figure 1: Comparison of the trajectory using RTM with the ground truth for the scaled Lorenz-63 system, where the blue star denotes the initial condition.
• Figure 2: Comparison of the trajectories using KLM and FDM with the ground truth for the scaled Lorenz-63 system, where the blue star denotes the initial condition.
• Figure 3: Comparisons of the trajectories with the approximated dynamics using three different methods and ground truth for the two-machine power system.
• Figure 4: Learned Lyapunov function and the corresponding region of attraction estimate using the random feature neural network dictionary functions for the reversed Van der Pol oscillator.
• Figure 5: RMSE of weights ($\mathcal{E}_{\operatorname{RMSE}}^{\text{W}}$) using RTM for reversed Van der Pol oscillator with different sampling frequencies.