Temporal fractional differential equations (FDEs) offer a powerful framework for modeling systems with memory and hereditary effects, commonly encountered in anomalous diffusion, viscoelasticity, and biological processes. However, their intrinsic non-locality presents significant challenges for conventional numerical methods. To address this, we propose a Global-Adaptive Physics-Informed Neural Network (GA-fPINN), an enhanced framework built upon standard PINNs. GA-fPINN integrates a deep neural network for discretizing fractional operators and embeds interpolation-based approximations to improve solution fidelity. Furthermore, it introduces three key adaptive technology: an uncertainty-aware loss function, a dual-parameter adaptive activation function, and a two-branch network architecture. Extensive evaluations on three representative FDE problems demonstrate that GA-fPINN consistently achieves higher accuracy and reduced computational cost compared to the baseline fPINN. These results highlight the framework’s potential for efficient and accurate modeling of complex fractional-order systems.

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GA-fPINN: Global-Adaptive Physics-Informed Neural Networks for Predicting the Nonlinear Dynamics of Temporal Fractional Order Differential Equations

  • Tailai Chen,
  • Ziyang Zhang,
  • Yuhan Yan,
  • Feifan Zhang

摘要

Temporal fractional differential equations (FDEs) offer a powerful framework for modeling systems with memory and hereditary effects, commonly encountered in anomalous diffusion, viscoelasticity, and biological processes. However, their intrinsic non-locality presents significant challenges for conventional numerical methods. To address this, we propose a Global-Adaptive Physics-Informed Neural Network (GA-fPINN), an enhanced framework built upon standard PINNs. GA-fPINN integrates a deep neural network for discretizing fractional operators and embeds interpolation-based approximations to improve solution fidelity. Furthermore, it introduces three key adaptive technology: an uncertainty-aware loss function, a dual-parameter adaptive activation function, and a two-branch network architecture. Extensive evaluations on three representative FDE problems demonstrate that GA-fPINN consistently achieves higher accuracy and reduced computational cost compared to the baseline fPINN. These results highlight the framework’s potential for efficient and accurate modeling of complex fractional-order systems.