Finite-element discretization methods are widely used in solving partial differential equations (PDEs), which describe dynamical systems governing traffic flow optimization. A mathematical model based on PDEs will be discretized using the finite-element method, with assumptions about traffic dynamics and network characteristics. This framework provides a robust theoretical basis for understanding and optimising traffic flow dynamics on South African roads using numerical methods. Further research should validate these findings with empirical data from real-world traffic scenarios. Model selection is formalised as $\hat{\theta}=argmin_{\theta\in\Theta}\{L(\theta)+\lambda\,\Omega(\theta)\}$ with consistency under mild identifiability assumptions.