Abstract: Taking the numerical solution of the circulation equation based on lifting line theory in the undergraduate Aerodynamics course as an example, this paper discusses the inherent logic underlying the evolution of solution paradigms for mathematical physics equations from series expansion to neural networks. This evolution unfolds along two dimensions: in terms of representation mode, it marks a leap from linear superposition to hierarchical composition; in terms of computational mechanism, it represents a progression from solving under rigid constraints to dynamic optimization guided by global physical information. The teaching case constructed in this paper demonstrates that, in the teaching practice of classical physics problems, introducing the comparative analysis of solution paradigms between neural networks and traditional series expansion helps students understand the universal law that "the macroscopic laws of complex systems can be characterized through the ordered assembly of simple primitives", and facilitates guiding students to focus on the inherent development logic of scientific research paradigms.