Levenberg–Marquardt supported mathematical modeling of Darcy–Forchheimer magnetized cross nanofluid flow with melting cylindrical surface and chemical processes using viscous dissipation

Many processes, such as the melting of permafrost, the solidification of magma, and the liquefying of frozen surface, are influenced by the melting process. The importance of understanding the melting procedures in fluid dynamics is established by freezing and melting of soil near the ground-based pipes’ heat exchanger loops. This work aims to study the impact of melting on the mass and heat transport characteristics of a Darcy-Forchheimer porous media in Cross-Nano fluidic flow in the direction of a cylindrical surface that is horizontally extended at the stagnation point of MHD. The method of heat transport is elaborated by viscous dissipation. Using the Cross nanofluidic model, parameters of Thermophoresis and Brownian motion studied. Furthermore, their influences on the solutal field are observed by chemical collaborations. The nonlinear coupled PDEs monitoring the fluid flow are transmuted into nonlinear ODEs by executing the proper similarity transformation. By means of unconventional Levenberg-Marquardt Backpropagation Scheme (LMB-Scheme) pooled with Neural Networks of Artificial Intelligence (NNs of AI) equations being numerically solved. To assess the accuracy of the model under various conditions, the efficiency of the anticipated NNs of AI with LMB-Scheme is evaluated through several metrics of performance, including plots of Regression Analysis (RA), analysis of Error Histogram (EA), and Mean Squared Error (MSE). A detailed computational investigation is performed to analyze the effects of key governing parameters on the velocity, temperature, and concentration fields of the unsteady MHD Carreau nanofluid model. The physical formulation incorporates thermal radiation, variable viscosity, and chemically reactive species, and the governing nonlinear partial differential equations are transformed into a system of ordinary differential equations using appropriate similarity transformations. The resulting system is solved through an artificial neural network framework trained with the Levenberg–Marquardt backpropagation scheme, and its accuracy is assessed using standard performance metrics including mean squared error, regression analysis, and convergence evaluation. The revised abstract therefore concludes by clearly outlining the problem formulation, computational strategy, and validation methodology adopted in this study.