The primary goal of this research is to present a novel computational intelligence approach of the AI-based Levenberg–Marquardt scheme under the influence of backpropagated neural network (LMS-BPNN) for optimizing MHD ternary hybrid nanofluid using Casson fluid over a porous shrinking sheet in the existence of thermal radiation ($Rd$) effects. The governing partial differential equations (PDEs) showing the Casson ternary hybrid nanofluid are converted into a system of ordinary differential equations (ODEs) with suitable transformations. The numerical data is constructed as a reference with bvp4c (MATLAB built-in function used to solve a system of ODEs) by varying Casson fluid parameters ($\beta$), magnetic field ($M$), porosity ($S$), nanoparticle concentrations (${\varphi }_1={\varphi }_2={\varphi }_3$), and thermal radiation ($Rd$) effects across all LMS-BPNN scenarios. The numerical data-sheet is divided into 80% of training, 10% of testing, and 10% of validation for LMS-BPNN are used to analyze the estimated solution and its assessment with a numerical solution using bvp4c is discussed. The efficiency and consistency of LMS-BPNN are confirmed via mean squared error (MSE) based fitness curves, regression analysis, correlation index ($R$) and error histogram. The results show that velocity decreases as $\beta$ grows, whereas velocity increase as $M$ increases. The concentrations of nanoparticles and thermal radiations have increasing effects on $\theta \left(0\right)$. To comprehend the dependability and correctness of the data gained from numerical simulations, error analysis is a key stage in every scientific inquiry. Error analysis is presented in terms of absolute error and it is noticed that the error between the numerical values and predicted values with AI is approximately ${10}^{-6}$. The error analysis reveals that the developed AI algorithm is consistent and reliable.

Highlighting the importance of artificial intelligence and machine learning approaches in engineering and fluid mechanics problems, especially in heat transfer applications is main goal of the presented article. With the advancement in Artificial Intelligence (AI) and Machine Learning (ML) techniques, the computational efficiency and accuracy of numerical results are enhanced. The theme of the study is to use machine learning techniques to examine the thermal analysis of MHD boundary layer flow of Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder embedded in a porous medium with heat source/sink and viscous dissipation effects. The considered base fluid is water ($H_{2}O$) and hybrid nanoparticles titanium oxide (${TiO}_2$) and Copper oxide ($CuO$). The governing flow equations are nonlinear PDEs. Non-similar system of PDEs are obtained with efficient conversion variables. The dimensionless PDEs are truncated using a local non-similarity approach up to third level and numerical solution is evaluated using MATLAB built-in-function bvp4c. Artificial Neural Networks (ANNs) simulation approach is used to trained the networks to predict the solution behavior. Thermal boundary layer improves with the enhancement in the value of $Rd$. The accuracy and reliability of ANNs predicted solution is addressed with computation of correlation index and residual analysis. The RMSE is evaluated [0.04892, 0.0007597, 0.0007596, 0.01546, 0.008871, 0.01686] for various scenarios. It is observed that when concentration of hybrid nanoparticles increases then thermal characteristics of the Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder.

The main idea of this investigation is to introduce an integrated intelligence approach that investigates the chemically reacting flow of non-Newtonian fluid with a backpropagation neural network (LMS-BPNN). The AI-based LMS-BPNN approach is utilized to obtain the optimal solution of an MHD flow of Eyring–Powell over a porous shrinking wedge with a heat source and nonlinear thermal radiation ($Rd$). The partial differential equations (PDEs) that define flow problems are transformed into a system of ordinary differential equations (ODEs) through efficient similarity variables. The reference solution is obtained with the bvp4c function by changing parameters as displayed in Scenarios 1–7. The label data are divided into three portions, i.e., 80% for training, 10% for testing, and 10% for validation. The label data are used to obtain the approximate solution using the activation function in LMS-BPNN within the MATLAB built-in command ‘nftool’. The consistency and uniformity of LMS-BPNN are supported by fitness curves based on the MSE, correlation index ($R$), regression analysis, and function fit. The best validation performance of LMS-BPNN is obtained at 462, 369, 642, 542, 215, 209, and 286 epochs with MSE values of 8.67 × 10⁻¹⁰, 1.64 × 10⁻⁹, 1.03 × 10⁻⁹, 302 9.35 × 10⁻¹⁰, 8.56 × 10⁻¹⁰, 1.08 × 10⁻⁹, and 6.97 × 10⁻¹⁰, respectively. It is noted that $f^{\prime }\left(\eta \right)$, $\theta \left(\eta \right),$ and $\varphi \left(\eta \right)$ satisfy the boundary conditions asymptotically for Scenarios 1–7 with LMS-BPNN. The dual solutions for flow performance outcomes (${Cf}_x$, ${Nu}_x$, and $S{h}_x$) are investigated with LMS-BPNN. It is concluded that when the magnetohydrodynamics increase ($M=0.01, 0.05, 0.1$), then the solution bifurcates at different critical values, i.e., ${\lambda }_c=-1.06329,-1.097,-1.17694$. The stability analysis is conducted using an LMS-BPNN approximation, involving the computation of eigenvalues for the flow problem. The deduction drawn is that the upper (first) branch solution remains stable, while the lower branch solution causes a disturbance in the flow and leads to instability. It is observed that the boundary layer thickness for the lower branch (second) solution is greater than the first solution. A comparison of numerical results and predicted solutions with LMS-BPNN is provided and they are found to be in good agreement.

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and mass transfer modeling in unsteady Maxwell fluid within a cylinder. Governing equations within a cylinder involve a system of coupled, nonlinear fractional partial differential equations (PDEs). A machine learning technique based on the Levenberg–Marquardt scheme with a backpropagation neural network (LMS-BPNN) is employed to evaluate the predicted solution of governing flow equations up to the required level of accuracy. The numerical data sheet is obtained using series solution approach Homotopy perturbation methods. The data sheet is divided into three portions i.e., $80\%$ is used for training, $10\%$ for validation, and $10\%$ for testing. The mean-squared error (MSE), error histograms, correlation coefficient (R), and function fitting are computed to examine the effectiveness and consistency of the proposed machine learning technique i.e., LMS-BPNN. Moreover, additional error metrics, such as R-squared, residual plots, and confidence intervals, are incorporated to provide a more comprehensive evaluation of model accuracy. The comparison of predicted solutions with LMS-BPNN and an approximate series solution are compared and the goodness of fit is found. The momentum boundary layer became higher and higher as there was an enhancement in the value of Caputo, fractional order α = 0.5 to α = 0.9. Higher thermal boundary layer (TBL) profiles were observed with the rising value of the heat source.