Model reduction of dynamical systems with a novel data-driven approach: The RC-HAVOK algorithm.

Area of Science:

• Dynamical Systems and Control Theory
• Data-Driven Modeling
• Nonlinear System Analysis

Background:

• The Koopman operator provides a powerful linear framework for analyzing nonlinear dynamical systems.
• Accurate and efficient approximation of the Koopman operator is crucial for model reduction and prediction.
• Existing data-driven methods face challenges in balancing accuracy and computational complexity.

Purpose of the Study:

• To introduce and evaluate the novel RC-HAVOK algorithm for data-driven Koopman operator approximation.
• To demonstrate the algorithm's effectiveness in reducing the dimensionality of the Koopman operator while minimizing error.
• To assess the performance of RC-HAVOK across diverse nonlinear dynamical systems.

Main Methods:

• The RC-HAVOK algorithm integrates Reservoir Computing (RC) with the Hankel Alternative View of Koopman (HAVOK) framework.
• This hybrid approach leverages RC's ability to generate high-dimensional state expansions and HAVOK's dimensionality reduction capabilities.
• The method is applied to Lorenz-like systems and systems with quadratic, cubic, hyperbolic tangent, and piecewise linear nonlinearities.

Main Results:

• RC-HAVOK successfully reduces the size of the linear Koopman operator with a lower error rate compared to other methods.
• The algorithm demonstrates high accuracy and feasibility across various nonlinear dynamical systems.
• Superior performance is observed, especially on standard Lorenz time series data, highlighting its practical utility.

Conclusions:

• The RC-HAVOK algorithm presents a significant advancement in data-driven Koopman operator approximation.
• It offers an effective and efficient approach for analyzing and modeling complex nonlinear dynamics.
• The method shows strong potential for applications in fields requiring accurate dynamical system identification from data.