Two methods to approximate the Koopman operator with a reservoir computer.

Area of Science:

• Dynamical Systems Analysis
• Data-Driven Modeling
• Operator Theory

Background:

• The Koopman operator framework is crucial for analyzing complex dynamical systems.
• Existing methods like extended dynamic mode decomposition (EDMD) often require extensive dictionary elements and nonlinear optimization.
• Efficient approximation of the Koopman operator necessitates well-chosen dictionary elements, typically obtained via complex training processes.

Purpose of the Study:

• To propose novel, efficient methods for training the dictionary in the Koopman operator framework.
• To leverage reservoir computing for dictionary training, replacing nonlinear optimization with linear convex optimization.
• To demonstrate the efficacy of these new methods in data reconstruction, prediction, and spectral analysis.

Main Methods:

• Development of two new dictionary training methods utilizing reservoir computers.
• Application of linear convex optimization techniques for dictionary training.
• Numerical validation across various data reconstruction and prediction tasks.

Main Results:

• The proposed reservoir computer-based methods successfully train the dictionary using only linear convex optimization.
• Demonstrated efficiency in data reconstruction and prediction tasks for dynamical systems.
• Successful computation of the Koopman operator spectrum using the trained dictionaries.

Conclusions:

• Reservoir computers offer a powerful and efficient alternative for training Koopman operator dictionaries.
• The proposed linear optimization methods simplify and enhance the application of the Koopman operator framework.
• These findings open new avenues for utilizing reservoir computing in data-driven system analysis.