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<p>The present study is associated to solve the Casson nanofluid mathematical model (CFMM) by designing a deep neural network (DNN) with 12 and 20 neurons using the hyperbolic tangent sigmoid activation function in both hidden layers. The governing equations are provided in the form of partial differential equations, which take the form of ordinary differential systems based on the similarity transformations. A construction of data is obtained through the Adam Bashforth optimizer, applied to lessen the mean square error by dividing the data into training, authentication, and testing with the statistics of 0.74, 0.14 and 0.12. The training of the data is performed by the Bayesian regularization scheme (BRS), which is an efficient scheme to solve the nonlinear systems. The nonlinear CFMM is categorized into the states of velocities, temperature, and concentration, while the numerical solutions are presented through the stochastic process based on DNN-BRS. The comparison of the results based on the obtained DNN-BRS and Runge-Kutta solver is also presented, which provides the confidence to solve the model. The overlapping of results is presented to signify the correctness of stochastic DNN-BRS and the negligible absolute error develops the worth of the proposed scheme.</p>

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Deep neural network modeling of casson nanofluid dynamics using bayesian regularization and hyperbolic tangent sigmoid activation

  • 期刊论文

  • 发表日期：

    2026年7月13日

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  • Zulqurnain Sabir,
  • M. A. Abdelkawy,
  • Umida Baltaeva,
  • Mustafa Bayram

摘要

The present study is associated to solve the Casson nanofluid mathematical model (CFMM) by designing a deep neural network (DNN) with 12 and 20 neurons using the hyperbolic tangent sigmoid activation function in both hidden layers. The governing equations are provided in the form of partial differential equations, which take the form of ordinary differential systems based on the similarity transformations. A construction of data is obtained through the Adam Bashforth optimizer, applied to lessen the mean square error by dividing the data into training, authentication, and testing with the statistics of 0.74, 0.14 and 0.12. The training of the data is performed by the Bayesian regularization scheme (BRS), which is an efficient scheme to solve the nonlinear systems. The nonlinear CFMM is categorized into the states of velocities, temperature, and concentration, while the numerical solutions are presented through the stochastic process based on DNN-BRS. The comparison of the results based on the obtained DNN-BRS and Runge-Kutta solver is also presented, which provides the confidence to solve the model. The overlapping of results is presented to signify the correctness of stochastic DNN-BRS and the negligible absolute error develops the worth of the proposed scheme.