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This study introduces a data-driven framework to uncover complex system equations from observational data. It accurately identifies dynamical information in stochastic systems without prior assumptions.

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Area of Science:

  • Complex Systems Analysis
  • Mathematical Physics
  • Data Science

Background:

  • Identifying governing equations of complex systems from data is a significant challenge.
  • Stochastic diffusion and jump-diffusion systems are prevalent in various scientific domains.

Purpose of the Study:

  • To develop a data-driven framework for discovering mathematical-physical equations of complex systems.
  • To identify dynamical information directly from observational time-series data.

Main Methods:

  • Utilizing the probability density function and Kramers-Moyal expansion.
  • Employing kernel density estimation with Fourier transform for Kramers-Moyal coefficient extraction.
  • Applying a data-driven sparse identification algorithm to reconstruct dynamic equations.

Main Results:

  • Successfully extracted Kramers-Moyal coefficients directly from system state variable time series.
  • Reconstructed underlying dynamic equations of stochastic systems using identified coefficients.
  • Demonstrated the framework's validity and accuracy in one- and two-dimensional examples.

Conclusions:

  • The proposed framework effectively identifies dynamical information and reconstructs governing equations for complex systems.
  • This data-driven approach bypasses the need for prior assumptions, offering a direct method from observational data.