Accurate modeling and control of fractional chaotic systems remain challenging due to their memory-dependent dynamics and sensitivity to initial conditions. In this paper, a topology-guided physics-informed neural network (PINN) framework is proposed to model, analyze, and control the fractional chaotic Lorenz system. The classical Lorenz system is generalized using the ψ-Hilfer fractional derivative. A PINN is developed to directly learn the fractional chaotic Lorenz system’s state trajectories while respecting its dynamics. Structure-preserving information is incorporated during training through a differentiable normalized pairwise-distance surrogate motivated by the Vietoris–Rips filtration, while Betti numbers, total persistence, persistence entropy, and Wasserstein distance are computed independently as post-training measures of topological fidelity. The PINN is evaluated for different parameters of the fractional chaotic Lorenz system. The numerical results demonstrate accurate trajectory reconstruction and enable post-training assessment of topological agreement, while a separately trained neural controller successfully stabilizes the fractional Lorenz system around the prescribed target equilibrium. Therefore, the proposed framework integrates fractional modeling, physics-informed learning, structure-guided reconstruction, persistent-homology analysis, and neural control within a unified two-stage methodology.