Finding nonlinear system equations and complex network structures from data: A sparse optimization approach

Area of Science:

• Dynamical Systems and Chaos Theory
• Network Science
• Machine Learning

Background:

• Many complex systems are measurable, but their underlying equations and dynamics remain unknown.
• Determining system equations from time-series data is a critical inverse problem.
• Sparse optimization offers a powerful approach to inferring system structure from data.

Purpose of the Study:

• To review recent advancements in using sparse optimization for discovering dynamical system equations.
• To discuss applications in predicting system transitions and inferring network topologies.
• To highlight the capabilities and limitations of sparse optimization in various dynamical systems.

Main Methods:

• Expanding system equations into finite power or Fourier series.
• Employing sparse optimization to determine expansion coefficients from time-series data.
• Comparing with traditional delay-coordinate embedding and mentioning machine learning frameworks.

Main Results:

• Successful identification of equations for stationary/nonstationary chaotic systems.
• Inference of complex network topologies (oscillator and social networks).
• Identification of partial differential equations for spatiotemporal systems.

Conclusions:

• Sparse optimization is a versatile tool for uncovering the governing equations of complex dynamical systems.
• Understanding the conditions under which sparse optimization succeeds or fails is crucial for its application.
• Data-driven, model-free approaches, including machine learning, are advancing prediction capabilities.