Primary information
ANN refers to a mathematical human/animal brain model. The brain of a human is made up of cells that are called neurons. The connection between these neurons generates a neural network. ANN is the recreation of the natural neural network, in which artificial neurons are linked exactly in the pattern of a genuine brain network. Furthermore, a neural network has produced significant developments in various areas of science, including recognizing images, natural language analysis, and automobiles.
Model for flow problem (ANN)
This mathematical model of an ANN receives one or more inputs and sums them to generate a single output. The sum is passed across a non-linear function, named an activation function. Figure 26 represents an artificial neural networking model with input values \({(x}_{1}, {x}_{2}, {x}_{3},\dots , {x}_{n}\)) and weights (\({w}_{1}, {w}_{2}, {x}_{1}, {w}_{3},\dots , {w}_{n})\). The total weight of the system is multiplied by all input values to obtain the input signal magnitude. A neuron receives numerous signals from its outer area and generates one output.
$$\left[ \begin{gathered} \sum\limits_{i = 1}^{n} {x_{i} } w_{i} = x_{1} w_{1} + x_{2} w_{2} + x_{3} w_{3} + ….x_{n} w_{n} , \hfill \\ y_{output} = f\left( {\sum\limits_{i = 1}^{n} {x_{i} } w_{i} } \right). \hfill \\ \end{gathered} \right].$$
(19)
Fig. 26
Figure 27 illustrates the overall workflow of the proposed methodology. Initially, parameter ranges are selected based on physical relevance and sensitivity analysis. The governing boundary value problem is solved using MATLAB bvp4c to generate velocity, temperature, concentration, and bioconvection profiles. These outputs, along with corresponding physical quantities (skin friction, Nusselt, Sherwood, and motile microorganism numbers), form the dataset for ANN training and testing. The multi-layer feed-forward ANN then predicts the physical responses for various parameter scenarios with high accuracy (R > 0.95), enabling efficient analysis and optimization without repeatedly solving the numerical model.
Fig. 27
Workflow diagram of this study.
ANN multi-layer feed forward method
In ANN, the technique of MLFF represents several hidden layers and different inputs with a single output. The connecting weights of the system give important network information. This network operates similarly in the human brain to enhance the quality of optimization and learning. The complete input equation for the jth hidden neuron’s layer is expressed as
$$y_{j} \left( x \right) = \sum\limits_{i = 1}^{l} {W1_{ji} x_{i} + a_{j} ,}$$
(20)
where, \(\left( {x_{i} ,a_{j} } \right)\) symbolize hidden layers, and \(x_{i}\) connected to \(a_{j}\) through \(W1_{ji}\). The network activation function is given as
$$z_{j} \left( x \right) = \frac{1}{{1 + e^{{ – y_{j} \left( x \right)}} }},$$
(21)
The formulation for the output equation is given as
$$O_{k} \left( x \right) = \sum\limits_{j = 1}^{m} {W2_{kj} z_{j} + b_{k} .}$$
(22)
The kth node is linked with the jth node through \(W2_{kj}\) and \(b_{k}\) used for the output layer. Errors with trials are being implemented to ensure stable learning convergence, hidden layer strength, along with input variables the present study, an Artificial Neural Network based on a multi-layer feed-forward structure was employed to predict the physical quantities obtained from the bvp4c numerical solution. The network consists of one input layer with 3 neurons (corresponding to the selected physical parameters), two hidden layers with 10 and 8 neurons, and one output layer for each target variable (skin friction, Nusselt number, Sherwood number, or motile microorganism density). Sigmoid activation functions were applied in the hidden layers, and a linear activation function was used in the output layer. Training was performed using the Levenberg–Marquardt (LM) backpropagation algorithm, which is widely used in engineering problems due to its fast convergence and low mean square error (MSE). With this architecture, the network achieved a correlation coefficient R > 0.95 and a minimum (MSE) of \(10^{ – 5}\) demonstrating high prediction accuracy.
Physical significance of tabular results
Table 3 presents the variation in different physical quantities, such as skin friction, Nusselt number, Sherwood number, and motile number, due to various physical parameters. Table 3 shows that skin friction decreases with increasing buoyancy ratio, modified Hartmann number, and Rayleigh number. This indicates that stronger buoyancy forces, enhanced magnetic effects, and intensified convection promote fluid flow along the surface, thereby reducing resistive shear stress at the wall. The results also reveal that higher Eckert numbers and heat relaxation parameters lower the Nusselt number, signifying reduced heat transfer. Larger Eckert numbers intensify viscous dissipation, increasing fluid temperature and weakening the wall temperature gradient, while higher heat relaxation slows the thermal response, further decreasing heat flux. Conversely, greater heat generation/absorption enhances the Nusselt number by strengthening the thermal gradient near the surface.
Table 3 Variation in various physical quantities.
For mass transfer, the Sherwood number increases with higher Schmidt and mass relaxation parameters due to reduced mass diffusivity and faster concentration response, respectively. However, stronger Brownian motion reduces Sherwood numbers by weakening concentration gradients. Regarding bioconvection, the motile density number decreases with higher Peclet and bioconvection parameters but increases with larger Lewis numbers, as stronger solute diffusivity relative to microorganism diffusivity enhances microorganism concentration near the wall.
Table 4 presents the coded and actual levels of the buoyancy ratio parameter \(N_{r}\), modified Hartmann number (Ha), and modified Rayleigh number \(N_{c}\), considered in the skin friction analysis. The three coded levels (− 1, 0, 1) represent the minimum, middle, and maximum values of each parameter, defining the parametric range for numerical investigation. This tabulation reflects the systematic variation of thermal buoyancy, magnetic field strength, and solutal convection to evaluate their influence on skin friction, serving as the input framework for subsequent analysis.
Table 4 Variation in skin friction with parameters.
Table 5 provides the coded and actual levels of the Eckert number (Ec), heat relaxation parameter, and heat generation parameter (Q) for the Nusselt number analysis. The coded levels (− 1, 0, 1) denote the minimum, middle, and maximum values of each parameter. Physically, Ec quantifies viscous dissipation effects, the heat relaxation parameter (\(\Gamma_{1}\)) accounts for thermal relaxation time, and Q represents internal heat generation. These values establish the input range for systematically examining their impact on heat transfer.
Table 5 Variation in Nusselt number with parameters.
Table 6 shows the coded and actual levels of the Schmidt number (Sc), mass relaxation parameter (\(\Gamma_{2}\)), and Brownian motion parameter (\(N_{b}\)), used in the Sherwood number analysis. Again, the coded levels (− 1, 0, 1) indicate the minimum, middle, and maximum values for each parameter. Here, Schmidt number Sc represents the ratio of momentum to mass diffusivity, \(\Gamma_{2}\) the mass relaxation parameter accounts for relaxation effects in mass transfer, and the Brownian motion parameter \(N_{b}\) characterizes the influence of nanoparticle movement. These ranges from the framework for analyzing concentration transport and overall mass transfer behavior.
Table 6 Variation in Sherwood number with parameters.
Table 7 presents the coded levels (− 1, 0, 1), which correspond to the minimum, middle, and maximum values assigned to each parameter. The Peclet number (Pe) reflects the relative significance of convection to diffusion, while the bioconvection Lewis number (Lb) describes the relationship between nanoparticle diffusion and microorganism movement. The non-dimensional bioconvection parameter represents the combined influence of motile microorganisms on the flow field. These ranges provide the basis for analyzing transport and flow behavior. Collectively, Tables 4–7 summarize the coded levels of various parameters, denoted by the symbols X, Y, and Z. The combined variations of these parameters are summarized in Table 8. The findings derived from the variations of three types of parameters and their respective effects as shown in Table 8. Here, the dataset for ANN training and testing was prepared by the authors from the numerical solutions obtained using MATLAB bvp4c solver for 21 parameter combinations. These combinations correspond to low, medium, and high levels coded as ( –1, 0, 1) of three parameters for each scenario: (Ha, \(N_{r}\), \(N_{c}\)) for velocity, (Ec, \(\Gamma_{1}\), Q) for temperature, (Sc, \(\Gamma_{2}\), \(N_{b}\)) for concentration, and (Pe, Lb, \(\varpi\)) for bioconvection. Each run produced outputs of skin friction, Nusselt number, Sherwood number, and motile number, which formed the ANN training dataset with 70% training, 15% validation, and 15% testing.
Table 7 Variation of motile number with parameters.Table 8 Responses of physical quantities with 21 runs.Heatmap analysis of parameters sensitivity
Figure 28 presents a sensitivity heat map to identify the impact of each physical parameter on the system’s outputs. It is evident that magnetic parameter Ha, the buoyancy ratio \(N_{r}\), and solutal parameters (Sc and \(\Gamma_{2}\)) have the most significant influence on velocity and mass transfer. Thermal parameters, including (Ec, Q, \(\Gamma_{1}\)) and predominantly control the Nusselt number. The bioconvection-related parameters Pe, Lb, and ω show a notable effect on the motile microorganism density number. This sensitivity analysis justifies the selection of parameter ranges for the ANN dataset, ensuring that all dominant physical effects are captured within the chosen variation limits.
Fig. 28
Sensitivity heatmap generated using Python 3.13 with Matplotlib 3.9.2 and Seaborn 0.13.2 (https://www.python.org/, https://matplotlib.org/; https://seaborn.pydata.org/).
The sensitivity heatmap in Fig. 28 was generated using Python (version 3.13) with the Matplotlib (version 3.9.2) and Seaborne (version 0.13.2) libraries. The heatmap was constructed by computing the parameter–quantity correlations from the trained PINN outputs and visualized with Seaborn’s heatmap function.