https://doi.org/10.1140/epjs/s11734-025-02005-z
Regular Article
Artificial Neural Network Modeling of Natural Convection Williamson Fluid Flow with Magnetic and Soret Effects in a Vertical Channel
Department of Mathematics, National Institute of Technology, Warangal, 506004, Warangal, Telangana, India
a
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Received:
18
April
2025
Accepted:
30
September
2025
Published online:
20
October
2025
Abstract
The study investigates the natural convection of Williamson fluid through a vertical channel, taking into account the effects of both the Soret phenomenon and the magnetic field while including a 
-ordered chemical reaction. The governing partial differential equations are turned into ordinary differential equations using a suitable transformation. Artificial neural networks (ANN) are employed to solve the flow problem. We utilize a feed-forward multilayered perceptron neural network to solve these ordinary differential equations. Further, the differential equation is approached with a trial solution consisting of two parts. The first part addresses initial or boundary conditions and remains parameter-free. Meanwhile, the second part involves training a feed-forward neural network to fulfill the requirements of the differential equation using the Adam optimizer technique. We assess the convergence and accuracy of our findings by comparing them with the Spectral Quasi Linearization Method (SQLM), resulting in satisfactory outcomes. The graphs demonstrate how changes in different parameters cause the velocity, concentration, and temperature curves. The outcome demonstrates that the axial velocity and concentration profiles decrease as the magnetic parameter value increases. Furthermore, the axial velocity and concentration trends amplify as the Hall parameter increases. Comparison with related optimizers like Stochastic Gradient Descent (SGD), Root Mean Square Propagation (RMSProp), and the Broyden Fletcher Goldfarb Shanno algorithm (LBFGS) was also explored in this study. The findings show that the Adam optimizer offers faster convergence and higher accuracy. The current study has significant applicability in industries such as food, pharmaceuticals, and petroleum, improving process optimization and product quality. Additionally, it advances fluid dynamics research by integrating machine learning techniques, paving the way for innovative solutions in complex fluid systems.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
