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Data-driven linearization of dynamical systems.

George Haller1, Bálint Kaszás1

  • 1Institute for Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092 Zurich, Switzerland.

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Summary
This summary is machine-generated.

Dynamic Mode Decomposition (DMD) is improved by a new justification for its applicability. Data-Driven Linearization (DDL) offers a more robust method for analyzing dynamical systems, outperforming existing techniques.

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Area of Science:

  • Dynamical Systems Theory
  • Data-Driven Modeling
  • Applied Mathematics

Background:

  • Dynamic Mode Decomposition (DMD) and its variants are widely used for linear modeling of dynamical systems from data.
  • Existing interpretations of DMD, particularly those based on the Koopman operator, have limitations and are based on restrictive assumptions.
  • A need exists to clarify the conditions under which DMD is applicable and to develop more robust methods.

Purpose of the Study:

  • To provide a rigorous justification for Dynamic Mode Decomposition (DMD) as a local, leading-order model for dominant system dynamics.
  • To develop a new, higher-order linearization algorithm, Data-Driven Linearization (DDL), for analyzing observable dynamics.
  • To demonstrate the superior performance of DDL compared to DMD and Extended DMD (EDMD).

Main Methods:

  • Developed a theoretical framework justifying DMD under conditions that hold with probability one for generic observables.
  • Constructed linearizing transformations for dominant dynamics within attracting slow spectral submanifolds (SSMs).
  • Introduced the Data-Driven Linearization (DDL) algorithm, a systematic, higher-order linearization technique.

Main Results:

  • Established conditions under which DMD provides a valid local, leading-order approximation of system dynamics.
  • The new DDL algorithm systematically linearizes observable dynamics within slow SSMs.
  • DDL demonstrated superior performance over DMD and EDMD on both numerical and experimental datasets.

Conclusions:

  • The study provides a robust theoretical foundation for DMD and introduces a more advanced method, DDL.
  • DDL offers a more accurate and systematic approach to linearization for dynamical systems analysis.
  • The findings advance the applicability and reliability of data-driven methods for understanding complex systems.