حل كننده‌هاي مبتني بر شبكه‌هاي عصبي آگاه از فيزيك مسئله براي برخي معادلات با مشتقات جزئي كسري

In this thesis, we present three novel physics-informed neural network (PINN) frameworks for solving forward an‎d inverse problems of fractional partial differential equations (FPDEs). First, the RBF-fPINN model is introduced, which employs radial basis function (RBF) networks as activation functions within a single hidden layer to significantly enhance solution accuracy. Various RBFs, including Gaussian, inverse quadratic, an‎d inverse multiquadric functions, are eva‎luated to identify the most effective configuration for solving FPDEs, particularly in complex 2D an‎d 3D domains. Next, for distributed-order FPDEs, the dPINN framework is developed to han‎dle high-dimensional, irregular, an‎d nonconvex domains. By integrating automatic differentiation with numerical discretization, dPINNs robustly solve both forward an‎d inverse problems, even in the presence of temporal singularities an‎d noisy initial data. This approach enables accurate identification of unknown parameters while maintaining stability in challenging computational settings, including advection-dominated transport scenarios. Finally, to address limitations of conventional MLP-based architectures in ill-posed problems, particularly fractional backward heat conduction, a novel framework based on the Kolmogorov–Arnold representation is proposed. Together, these methods demonstrate enhanced accuracy, robustness, an‎d adaptability of physics-informed neural networks for complex fractional models, offering a promising computational tool for engineering an‎d physical sciences applications. Numerical results show the efficiency of the proposed schemes.

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هادي روحاني , سجاد لكزيان , بهزاد نعمتي سراي