Extended Dynamic Mode Decomposition with Invertible Dictionary Learning.

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning for Scientific Modeling
• Nonlinear System Analysis

Background:

• The Koopman operator offers a global linearization for nonlinear dynamical systems.
• Data-driven modeling requires invertible observables for accurate state reconstruction.
• Current methods achieve only lossy or limited nonlinear reconstruction.

Purpose of the Study:

• To develop a data-driven modeling approach for nonlinear dynamical systems that allows for nonlinear and lossless state reconstruction.
• To extend the capabilities of Extended Dynamic Mode Decomposition (EDMD) for improved system estimation and control.
• To address the invertibility problem in Koopman operator-based modeling.

Main Methods:

• Proposed Extended Dynamic Mode Decomposition with Invertible Dictionary Learning (EDMD-IDL).
• Incorporated Invertible Neural Networks (INNs) for explicit inverse dictionary functions.
• Developed an iterative algorithm combining gradient descent and EDMD for Koopman operator approximation.

Main Results:

• Achieved nonlinear and lossless reconstruction of system states.
• Demonstrated accurate long-term predictions for canonical nonlinear systems with only initial state data.
• Showcased superior performance compared to existing EDMD-based methods.
• Successfully reconstructed the Kármán vortex street phenomenon in fluid dynamics using Proper Orthogonal Decomposition.

Conclusions:

• EDMD-IDL provides a novel paradigm for finite-dimensional approximation of the Koopman operator.
• The method enables accurate, data-driven modeling and prediction of complex nonlinear systems.
• Potential for extension to high-dimensional systems, including applications in fluid dynamics.