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<p>Fractional differential equations (FDEs) have been used extensively to model systems with memory and non-local dynamics, but the high computational cost of evaluation of a fractional derivative means that their numerical solution is difficult to calculate. Classical methods like finite difference have the disadvantage of needing to store the entire history of the solutions, whereas Physics-Informed Neural Networks (PINNs) have the drawback of being unstable and high cost to train, and standalone Monte Carlo (MC) method exhibit slow convergence and large variance. To overcome these limitations, this study present a hybrid PINN-MC that will integrate MC sampling in the PINN architecture to effectively approximate the Caputo fractional derivative and minimize memory requirements and enhance computational efficiency. Numerical results of nonlinear fractional models such as decay or growth equation, damping equation, predator-prey type model, and Lotka-Volterra model indicate that the proposed method has high accuracy and convergence relative to the standard MC method, with low approximation error and high stability.</p>

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Computational modeling of Monte Carlo–enhanced PINNs for fractional order differential models with memory effects

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月1日

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  • Hamad Jan,
  • Muhammad Sulaiman,
  • Tauqeer Ahmad,
  • Ghaylen Laouini,
  • Izaz Ur Rahman,
  • Mohammed Abdullah Salman

摘要

Fractional differential equations (FDEs) have been used extensively to model systems with memory and non-local dynamics, but the high computational cost of evaluation of a fractional derivative means that their numerical solution is difficult to calculate. Classical methods like finite difference have the disadvantage of needing to store the entire history of the solutions, whereas Physics-Informed Neural Networks (PINNs) have the drawback of being unstable and high cost to train, and standalone Monte Carlo (MC) method exhibit slow convergence and large variance. To overcome these limitations, this study present a hybrid PINN-MC that will integrate MC sampling in the PINN architecture to effectively approximate the Caputo fractional derivative and minimize memory requirements and enhance computational efficiency. Numerical results of nonlinear fractional models such as decay or growth equation, damping equation, predator-prey type model, and Lotka-Volterra model indicate that the proposed method has high accuracy and convergence relative to the standard MC method, with low approximation error and high stability.