Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the...

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning Applications
• Scientific Computing

Background:

• Data-driven methods like Dynamic Mode Decomposition (DMD) and Extended DMD (EDMD) are increasingly used for Koopman operator approximation.
• EDMD enhances accuracy by using a flexible dictionary of observables, but requires careful selection for high-dimensional, nonlinear systems.
• Existing EDMD methods face challenges in optimal dictionary selection for complex dynamical systems.

Purpose of the Study:

• To develop an improved EDMD algorithm using machine learning for adaptive dictionary selection.
• To enhance the accuracy and efficiency of Koopman operator approximation in complex systems.
• To reduce the number of required dictionary terms for a given accuracy.

Main Methods:

• An iterative approximation algorithm coupling EDMD with a trainable dictionary represented by an artificial neural network was developed.
• The algorithm was tested using the Duffing oscillator and the Kuramoto-Sivashinsky partial differential equation.
• A machine learning approach was used to adapt the dictionary of observables automatically.

Main Results:

• The proposed algorithm effectively and efficiently adapts the trainable dictionary to specific problems.
• Good reconstruction accuracy was achieved without the need for a priori fixed dictionary selection.
• Fewer dictionary terms were required to reach a given accuracy compared to traditional EDMD with fixed dictionaries.

Conclusions:

• The developed algorithm alleviates a key limitation of EDMD, enhancing its practical applicability.
• This machine learning-enhanced EDMD approach broadens the utility of the Koopman framework for complex dynamical systems.
• The method demonstrates superior performance in adapting dictionaries for accurate Koopman operator approximation.