Numerical analysis and comparison of fractional physics-informed neural networks in unsaturated flow process

This paper investigates the application of deep learning techniques to unsaturated flow problems, integrating the fractional Richards model with a fractional physics-informed neural network. We introduce an innovative Hermite neural network solver that enhances the model’s performance and accuracy in addressing complex flow challenges through an attention mechanism. Systematic comparisons with conventional fractional neural network solvers demonstrate that our Hermite interpolation-based network significantly surpasses others in computational precision and efficiency. To further validate our proposed method, we conduct comprehensive numerical simulations of two-dimensional scenarios utilizing the attention-enhanced network. The results from these simulations illustrate the robustness of our approach and highlight its potential for accurately capturing flow characteristics and dynamic behaviors within the model. This study also offers new insights and methodologies for leveraging deep learning in unsaturated flow research, establishing a solid foundation for future investigations and practical engineering applications.