Dynamic mode decomposition for multiscale nonlinear physics.

Area of Science:

• Complex Systems Analysis
• Data-Driven Modeling
• Dynamical Systems Theory

Background:

• Multiscale systems present challenges in analysis due to overlapping temporal and spatial scales.
• Existing multiresolution analysis (MRA) methods may struggle with simultaneous spatial and temporal coherencies.
• Accurate separation and reconstruction of system components are crucial for understanding and prediction.

Purpose of the Study:

• To develop a data-driven method for decomposing complex, multiscale systems into their constituent timescale components.
• To provide a robust approach that accounts for both spatial and temporal coherencies.
• To enable faithful reconstruction and short-term prediction of individual system components.

Main Methods:

• A recursive implementation of dynamic mode decomposition (DMD) is employed.
• Local linear models are constructed from windowed data subsets.
• Spectral clustering on eigenvalues identifies dominant timescales, yielding time series for each component.

Main Results:

• The proposed method successfully separates multiscale systems into distinct timescale components.
• Components sum to a faithful reconstruction of the original input signal.
• The approach demonstrates robustness to scale overlap and provides local dynamics for prediction.

Conclusions:

• The generalized multi-resolution dynamic mode decomposition (mrDMD) offers a powerful tool for analyzing diverse multiscale systems.
• The method provides accurate component isolation and enables forecasting applications.
• This technique advances the field of multiresolution analysis with its integrated spatial-temporal approach.