Thermal transport plays a crucial role in high-temperature engineering and energy systems where radiative heat transfer becomes a dominant mechanism influencing temperature distribution and system performance. In this paper, a comprehensive investigation is conducted on radiative thermal transport in magnetohydrodynamic (MHD) flow of Carreau nanofluids over a rectangular frame incorporating both active and passive controls. The novelty of this study lies in the simultaneous integration of non-Newtonian Carreau nanofluid rheology, nanoparticle shape dynamics, magnetic field effects, thermal radiation and an artificial neural network (ANN)-assisted predictive framework, which has not been previously examined in a unified configuration. The governing mathematical model describing momentum, thermal, and concentration transport is formulated using partial differential equations and transformed into a system of ordinary differential equations for numerical computation. A hybrid computational method is employed in which numerical solutions obtained via the bvp4c solver scheme and then the specified data is utilized to train a Levenberg-Marquardt based neural network. The results demonstrate that thermal radiation intensifies the temperature distribution for spherical, cylindrical, and platelet nanoparticle shapes, which highlights its dominant role in radiative heat transport mechanisms. Brownian motion is observed to reduce nanoparticle concentration, whereas thermophoresis significantly enhances mass transport in both active and passive control configurations. These findings provide new physical insights into thermal transport mechanisms in Carreau nanofluids and offer a reliable predictive tool for advanced thermal management and high-temperature engineering applications.