Physics-Informed Neural Networks (PINN) represent a powerful approach in machine learning that enables the solution of forward, inverse, and parameter identification problems related to models governed by fractional differential equations. This is achieved by incorporating residuals of operator equations, boundary, and initial conditions into the objective function during training. The proposed approach focuses on an adaptive inverse fractal-oriented PINN designed for modeling heat and moisture transfer in capillary-porous materials with a fractal structure and identifying unknown fractional parameters. The core idea is to first construct a fractal neural network for solving the forward problem and then extend its application by transforming fractional derivative orders into trainable variables for optimization. Additionally, synthetic data are incorporated into the objective function to ensure the necessary conditions for solving the identification problem. To ensure that the approximate solution accurately reproduces the physical behavior of the system, the components of the loss function such as deviations from synthetic data, initial and boundary conditions, and residuals of differential equations are adaptively weighted at each training epoch. Similarly, the gradients of trainable parameters are scaled accordingly during the training process. To confirm the effectiveness and reliability of this approach, several examples obtained using the developed software are presented. These examples illustrate its application in various specific scenarios and demonstrate the ability of the adaptive fractal PINN to successfully solve heat and mass transfer problems in fractal capillary-porous structures, as well as accurately identify fractional parameters.
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