The purpose of this study is to investigate the unsteady Caputo fractional fluid flow in the annular region of a vertical cylinder, incorporating heat supply or loss under natural convection and the influence of Hall current along a radial magnetic field. Fractional time derivatives of the Caputo type replace traditional integer-order derivatives to more accurately capture memory effects, anomalous diffusion, and sub-diffusive transport more accurately. To solve the resulting fractional model, a physics-informed neural network (PINN) framework is employed as a mesh-free alternative to conventional numerical techniques. By embedding the initial and boundary conditions, along with the governing fractional partial differential equations, directly into the loss function, the PINN effectively approximates both velocity and temperature fields. Automatic differentiation and the expressive capability of deep neural networks facilitate the treatment of concentric geometries and nonlocal fractional operators. The predicted results show strong agreement with existing literature, validating the accuracy of the proposed approach. Additionally, the results are computed and compared in terms of geometric aspects with small (λ=4) and large radial gaps (λ=10). Forλ=4, the curves are exactly parabolic, whereasλ=10, the profiles are attaining an asymptotic approach due to the ignorance of curvature effects for large r. The magnitude of the steady-state velocity at r = 4 increases by 10.29%, 52.33%, and 100% of its maximum velocity for the corresponding increment of α=0.1,0.5and0.9. Similarly, the temperature reaches 6.89%, 9.39%, and 100% of its maximum temperature for the same increments of α=0.1,0.5and0.9.