The roll coating process is broadly employed in the manufacturing of wallpapers, protection of fabrics and metals, wrapping, adhesive tapes, x-ray and photographic films, books and magazines, beautification, magnetic records, film foils, coated paper, etc. This study proposes a new framework for analyzing non-Newtonian fluid flow between co-rotating rolls at identical speed and size. The framework combines analytical, numerical, and computational methods powered by artificial neural networks. A key aspect of the model is the incorporation of temperature-dependent viscosity, allowing us to investigate its theoretical influence on various flow characteristics and relevant engineering parameters. To achieve this, we derive non-dimensionalized mass and momentum balance equations using appropriate transformations and lubrication approximation theory. The analytic expression for velocity distribution, temperature, pressure gradient, pressure fields, and flow rate is achieved by utilizing the perturbation method. The numerical solutions using the collocation method based on Hermite functions and the boundary value problem built-in method are also obtained. After deriving these expressions, we calculate engineering quantities including the Nusselt number, streamline, power input needed to drive both cylinders, and the roll separation force. The impacts of emerging parameters on all quantities of interest are illustrated using graphs and tables. It is interesting to mention that an increase in the non-Newtonian parameter increases the velocity but in the increase in the Vogel viscosity parameter, the velocity decreases. Furthermore, the correctness of the present work is observed by comparing analytic, numeric solutions and previously published work, and observed good agreement. To obtain approximate solutions for various flow scenarios within the proposed model, we employ a supervised neural network solver with Levenberg–Marquardt backpropagation (LMBP-SNNs) for testing, validation, and training. This approach utilizes the mean squared error (MSE) to adjust the network parameters. The efficiency of the proposed LMBP-SNN solver is validated through a combination of comparative analyses, performance studies based on MSE outputs, and visualizations of regression errors. The performance on MSE has been shown for the velocity profiles of the developed model about 9.174 × 10−12, 4.1029 × 10−12, 4.5997 × 10−12, and 2.8300 × 10−13. This study addresses a gap in the existing literature by exploring the rheological properties of the Eyring–Powell fluid model and integrating numerical methods along with machine learning techniques in the forward roll coating process.
