Dynamical and Numerical Analysis of Nonlinear Diffusion Models Using Lie Symmetry and PINN

We perform a Lie symmetry classification of a two-dimensional nonlinear diffusion-reaction equation with anisotropic diffusion and reaction terms. Using the obtained Lie symmetries, we derive the corresponding reduced equations and construct several exact solutions, including explicit and implicit forms, as well as approximate power-type solutions. Representative cases from the symmetry classification are selected, and the derived exact solutions are used to design consistent initial and boundary conditions. A PINN-based approximation is then used to compute numerical solutions, which are compared against the exact solutions through error metrics. Heatmaps illustrate the agreement of the trained PINN model. Finally, the reduced ODEs are examined via equilibria, local stability, and phase portraits. The qualitative changes induced by parameter variations are discussed, clarifying how parameter choices affect the reduced and numerical solutions.