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Neural network representation of the probability density function of diffusion processes.

Wayne Isaac T Uy1, Mircea D Grigoriu1

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Chaos (Woodbury, N.Y.)
|October 2, 2020
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Physics-informed neural networks characterize dynamical systems in random environments by approximating probability density functions. This approach offers insights into system behavior under various random forcings.

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Area of Science:

  • Dynamical systems theory
  • Computational physics
  • Machine learning

Background:

  • Dynamical systems in random environments are challenging to analyze.
  • Characterizing system states requires approximating probability distributions.
  • Traditional methods struggle with complex stochastic differential equations.

Purpose of the Study:

  • To develop physics-informed neural networks (PINNs) for characterizing dynamical systems.
  • To approximate the probability density function (pdf) or characteristic function (chf) of system states.
  • To investigate the efficacy of PINNs for systems governed by Fokker-Planck or integro-differential equations.

Main Methods:

  • Utilizing PINNs to approximate the pdf or chf of system states.
  • Solving Fokker-Planck and integro-differential equations using neural networks.
  • Analyzing the advantages and disadvantages of different equation formulations.
  • Incorporating prior knowledge of dynamical systems to simplify network architecture.

Main Results:

  • PINNs successfully approximate solutions for complex partial integro-differential and systems of partial differential equations.
  • Neural network solutions for Fokker-Planck and chf equations yield comparable state probability density functions.
  • The developed method effectively characterizes system behavior under diverse random forcings.

Conclusions:

  • PINNs provide a robust framework for analyzing dynamical systems in random environments.
  • Approximating either the pdf or chf using PINNs is a viable strategy.
  • Prior information integration enhances PINN efficiency for stochastic system analysis.