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.

To enhance the productivity of heat exchangers, a new type of heat transfer fluid called hybrid nanofluid was characterized. An abundance of industrial and technological processes depends on heat transfer through these fluid flows, whether in laminar or turbulent conditions. The thermal resistance of the fluid based on a heat transfer system might be increased to improve a significant number of applications. Therefore, the present investigation, establishing a framework for differential equations to examine the influence of thermal radiation and heat source/sink on magnetohydrodynamics (MHD) Marangoni convection flow of hybrid nanofluid (HNF) in a Saturated Porous medium. By introducing new similarity variables, the problem of Marangoni convection is simplified to account for the effects of radiation and porous medium. This study’s originality is the combined impact of the heat source and porous material. The governing nonlinear partial differential equations for momentum and energy are transformed into ordinary differential equations by applying the required similarity adjustment. The shooting strategy is used to achieve numerical solutions for certain modeled ordinary differential equations. Excellent agreement with the shooting technique in the MATLAB (bvp4c) code is demonstrated by a comprehensive numerical comparison conducted for a variety of variables. Investigations are conducted into specific physical characteristics related to temperature and velocity profiles. For several factors, the results are tabulated and graphically analyzed. The velocity plots are increasing with the rise in the Stretching parameter, and volume quantity of copper, while for large values of marangoni parameter, porosity parameter, and Magnetic parameter the opposite trend shows. Larger volume concentrations of copper, larger values of heat source parameter, thermal radiation, and stretching parameter features all increase the temperature pattern, while opposite trends shows for porosity components, magnetic, and marangoni parameter. The graph of streamlines and isotherms are also plotted. The numerical results were compared with previous published work.