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Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries.

Syuzanna Sargsyan1, Steven L Brunton2, J Nathan Kutz1

  • 1Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA.

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Summary
This summary is machine-generated.

This study introduces sparse sampling and dimensionality reduction for modeling nonlinear dynamical systems. The method efficiently captures system dynamics and bifurcations using fewer data points, improving computational speed.

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Area of Science:

  • Nonlinear Dynamics
  • Computational Physics
  • Data Science

Background:

  • Characterizing nonlinear dynamical systems is computationally intensive.
  • Traditional methods struggle with high-dimensional data and complex dynamics.
  • Bifurcation analysis requires accurate modeling of system behavior across parameter ranges.

Purpose of the Study:

  • To develop a sparse sampling and dimensionality reduction technique for modeling nonlinear dynamical systems.
  • To enable efficient characterization of system dynamics, stability, and bifurcations.
  • To reduce computational cost and memory requirements for analyzing complex systems.

Main Methods:

  • Constructing modal libraries using Proper Orthogonal Decomposition (POD) to identify dominant structures.
  • Employing the Discrete Empirical Interpolation Method (DEIM) for approximating nonlinear terms from sparse grid points.
  • Developing local reduced-order models based on empirical interpolation points for physical interpretability.

Main Results:

  • Identified effective sparse sensing locations for characterizing dynamics and bifurcations.
  • Demonstrated orders-of-magnitude improvements in computational speed and memory usage.
  • Successfully classified and reconstructed dynamic bifurcation regimes in the complex Ginzburg-Landau equation.
  • Showcased the superiority of nonlinear point measurements over linear measurements in the presence of sensor noise.

Conclusions:

  • Sparse sampling and DEIM provide an efficient framework for modeling nonlinear dynamical systems.
  • The method facilitates physically interpretable reduced-order models and enhances computational performance.
  • This approach is effective for analyzing complex phenomena like bifurcations and is robust to sensor noise.