Approximation of translation invariant Koopman operators for coupled non-linear systems

Area of Science:

• Physics
• Applied Mathematics
• Data Science

Background:

• Many physical systems possess translational invariance, where physical laws remain constant across different spatial positions.
• Accurate modeling of non-linear dynamical systems using data-driven methods requires incorporating these physical symmetries.
• Existing dynamic mode decomposition (DMD) techniques may not adequately preserve translational invariance.

Purpose of the Study:

• To introduce a novel method, translation invariant extended dynamic mode decomposition (tieDMD), for analyzing coupled non-linear systems on periodic domains.
• To ensure data-driven approximations of the Koopman operator respect translational symmetry.
• To evaluate the performance of tieDMD against existing DMD variants.

Main Methods:

• Exploiting the block-diagonal structure of the Koopman operator in Fourier space to achieve translation invariance.
• Applying variants of tieDMD to datasets from the phase-diffusion, Burgers, Korteweg-de Vries, and FitzHugh-Nagumo equations.
• Comparing the data reconstruction capabilities of tieDMD with traditional linear and non-linear DMD methods.

Main Results:

• tieDMD demonstrated superior data reconstruction capabilities across various non-linear partial differential equations.
• The method effectively leverages Fourier space to enforce translational symmetry in Koopman operator approximations.
• Consistent improvements in reconstruction accuracy were observed for all tested datasets.

Conclusions:

• tieDMD offers a physically informed approach for modeling non-linear dynamical systems with translational invariance.
• The method provides a significant advancement in data-driven model accuracy and reliability.
• tieDMD is a valuable tool for analyzing complex physical systems exhibiting symmetries.