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Area of Science:

  • Complex systems science
  • Data-driven modeling
  • Stochastic processes

Background:

  • Data-driven methods are increasingly vital for understanding complex phenomena.
  • Discovering dynamical behaviors in stochastic systems from data remains challenging.
  • Existing methods may not fully capture the intricacies of systems with noise.

Purpose of the Study:

  • To propose a novel framework for detecting dynamical behaviors in stochastic dynamical systems using observational data.
  • To enable the calculation of the maximum likelihood transition path from empirical data.
  • To validate the framework's efficacy on systems with different types of Gaussian noise.

Main Methods:

  • Utilizing the Kramers-Moyal formula to connect sample path data with system coefficients.
  • Employing the extended sparse identification of nonlinear dynamics (SINDy) for coefficient estimation.
  • Calculating the maximum likelihood transition path based on estimated system dynamics.

Main Results:

  • The proposed framework successfully links sample path data to system coefficients.
  • Accurate estimation of coefficients for stochastic dynamical systems was achieved.
  • The maximum likelihood transition path was effectively calculated for tested systems.
  • Framework validity demonstrated by reproducing known behaviors in systems with additive and multiplicative Gaussian noise.

Conclusions:

  • The developed framework offers a robust approach for analyzing stochastic dynamical systems from data.
  • This method provides a powerful tool for uncovering hidden dynamical behaviors, including critical transition paths.
  • The findings have implications for various scientific fields relying on the analysis of complex systems.