Multi-tiered AI-driven symbolic reasoning for solving the 2D linear advection equation

Abstract This study examines the two-dimensional linear advection equation, a mathematical model that characterizes the temporal evolution of a physical field within a two-dimensional spatial domain. This equation is frequently employed to simulate transport phenomena in fluid dynamics and material transfer processes. By combining deep learning techniques with symbolic reasoning, we propose sophisticated neural network architectures featuring one, two, and three hidden layers. Each architecture uses a wider variety of activation functions. Consequently, exact analytical solutions characterized by multiple arbitrary functions are derived. Furthermore, the computational software Maple is employed to determine suitable parameters and generate visualizations, including two-dimensional plots, three-dimensional plots, and density maps, thereby effectively demonstrating the various analytical solutions derived. This innovative approach offers a strong foundation for solving equations in fields like fluid dynamics and computational meteorology, demonstrating the enormous promise of artificial intelligence in mathematical physics.