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This study approximates stochastic differential equations using spectral density. The method infers underlying laws from noisy data, applicable to linear processes and time-series analysis.

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Area of Science:

  • Applied Mathematics
  • Statistical Physics
  • Information Theory

Background:

  • Stochastic differential equations (SDEs) are crucial for modeling complex systems where precise equations are unknown or computationally expensive.
  • Examples include Brownian motion and macroscopic physical processes where underlying dynamics are not easily deduced from first principles.
  • Inferring these dynamics from observational data is a significant challenge in many scientific fields.

Purpose of the Study:

  • To develop a method for approximating the underlying laws of stochastic processes using their spectral density.
  • To demonstrate the inference of these laws from potentially noisy and incomplete measurements.
  • To apply the developed reconstruction algorithm to linear and autonomous time-series and spatiotemporal processes.

Main Methods:

  • Approximation of stochastic process dynamics via spectral density.
  • Inference of spectral density from noisy and incomplete time-series data.
  • Application of Information Field Theory principles for inverse problem-solving.
  • Reconstruction algorithm tested on linear, autonomous time-series and spatiotemporal data.

Main Results:

  • Successfully approximated the underlying laws of stochastic processes using spectral density.
  • Demonstrated the ability to infer these laws even with limited and noisy measurement data.
  • Validated the reconstruction algorithm's effectiveness on both time-series and spatiotemporal examples.

Conclusions:

  • Spectral density provides a viable approach to approximate unknown laws governing stochastic processes.
  • The developed method offers a robust way to infer system dynamics from empirical data.
  • This work contributes a novel technique for analyzing complex dynamical systems in science and industry.