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Spatiotemporal system reconstruction using Fourier spectral operators and structure selection techniques.

Daolin Xu¹, Omid Khanmohamadi

¹School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798.

Chaos (Woodbury, N.Y.)

|January 7, 2009

Summary

This study introduces a novel spectral method for reconstructing complex nonlinear partial differential equations from time series data. The technique accurately identifies system dynamics and simplifies models, even with limited data.

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Area of Science:

Applied Mathematics
Computational Science
Nonlinear Dynamics

Background:

Reconstructing complex spatiotemporal systems from observational data is challenging.
Existing methods may struggle with high nonlinearity and arbitrary derivative orders.

Purpose of the Study:

To develop a robust technique for identifying and reconstructing nonlinear partial differential equations from time series.
To create parsimonious models by eliminating redundant parameters.

Main Methods:

Trigonometric spectral methods combined with structure selection.
Fourier spectral differentiation operators for system identification.
Orthogonal decomposition for parameter reduction.

Main Results:

The proposed technique successfully reconstructs spatiotemporal systems.
Demonstrated superior accuracy and robustness, even with sparse data.
Effective for highly stiff reaction-diffusion systems like the Kuramoto-Sivashinsky equation.

Conclusions:

The spectral method offers an accurate and robust approach for nonlinear system identification.
This technique facilitates the creation of simplified, yet effective, mathematical models.
Applicable to a range of nonlinear partial differential equations and complex systems.