Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization.

Area of Science:

• Data-driven modeling
• Nonlinear dynamical systems
• Scientific computing

Background:

• Modern data-driven dynamical systems research faces challenges in high dimensionality, unknown dynamics, and nonlinearity.
• Dynamic Mode Decomposition (DMD) is a key tool for high-dimensional systems but is sensitive to nonlinearity.
• Sparse identification of nonlinear dynamics handles nonlinearity but is limited to low-dimensional systems.

Purpose of the Study:

• To develop a novel kernel method for learning interpretable data-driven models for high-dimensional, nonlinear systems.
• To address the limitations of existing methods like DMD in handling strong nonlinearities and high dimensionality.
• To enable the separation of linear and nonlinear effects in complex dynamical systems.

Main Methods:

• Kernel regression on a sparse dictionary of influential samples.
• Application of a kernel method to high-dimensional data.
• Incorporation of partial knowledge of system physics into the model.

Main Results:

• The kernel method efficiently handles high-dimensional data.
• The approach successfully separates linear model contributions from nonlinear terms.
• Demonstrated effectiveness on data from various nonlinear ordinary and partial differential equations.

Conclusions:

• The proposed kernel method provides a flexible and interpretable framework for modeling high-dimensional nonlinear systems.
• This approach overcomes limitations of traditional DMD and sparse identification methods.
• The framework supports diverse engineering tasks including model reduction, prediction, and control.