Mathematical and computational modeling of biological systems: advances and perspectives

Nomenclature

1.   Introduction

Neural networks are widely utilized, having applications in enterprise planning, financial operations, trading, product maintenance and business analytics. Neural networks are also widely used in corporate applications such as marketing exploration and forecasting, risk assessment, and fraud detection. Backpropagation is a popular learning process in input multilayer neural networks. This approach has lately been employed by researchers to investigate mass and heat transfer characteristics, as well as non-Newtonian fluid flow systems. In 1974 Paul Werbos was the foremost person to design the backpropagation technique. This method was revived and improved upon by Rumelhart and Parker ^[1] after Paul Werbos's discovery. An artificial neural network (ANN) is an innovative new artificial intelligence method. ANNs can exploit a variety of disciplines throughout the learning development depending on the data that flows through the network, whether it initiates from inside or outside of it. To enhance the multilayer perceptron (MLP) network's performance, an ANN uses the back propagation technique for simultaneous training. It is the most widely used, easy-to-learn, and successful model for complex multilayer networks. A unique convergent consistency technique for ANNs is the Levenberg-Marquardt technique (LMT), which provides a numerical solution to a range of fluid flow issues. Various researchers have lately utilized this technique to analyze non-Newtonian fluid flow systems and their properties related to mass and heat transfer. In their study, Ajed et al. ^[2] suggested using neural network design to inspect the heat transfer and MHD in a two-phase paradigm of nanofluid flow when thermal energy exists. Over a stretched sheet, Shoaib et al. ^[3] examined the numerical analysis for rotational MHD nanofluid flow hybrids with heat radiation using neural networks. Utilizing the back-propagated neural networks with Levenberg-Marquard system (BNN-LMS), Khan et al. ^[4] investigated the heat transfer across two porous parallel surfaces of stable nanofluids utilizing thermophoresis and Brownian effects. Using a backpropagation Levenberg-Marquardt, Sabir et al. ^[5] studied methods of computational intelligence to clarify the Emden-Fowler paradigm's third-order nonlinear scheme. Shafiq et al. ^[6] explored the modeling of Soret and Dufour's convective heat transfer in nanofluid flow through a moving needle with an artificial neural network. Shoaib et al. ^[7] exercised the technique of Levenberg–Marquardt back propagation with neural networks (TLMB-NN) and a stochastic numerical approach via ANNs to study the consequences of heat transfer on Maxwell flow of nanofluid above a vertical moving surface with MHD. Intelligent networks were utilized by Ajed et al. ^[8] to study MHD flow of fluid in a thermally stratified medium across coaxial extensible spinning disks. Sindhu et al. ^[9] investigated the reliability study of generalized exponential distribution based on inverse power law using an artificial neural network with Bayesian regularization.

To compare the performance of different neural network models for robustness verification, we need to consider various factors such as the type of verification, the size and complexity of the network, and the robustness metric used ^[10].

Types of verification:

ⅰ) Point-wise verification

ⅱ) Partition-based verification

Alternative neural network models:

ⅰ) Deep neural networks (DNNs)

ⅱ) Convolutional neural networks (CNNs)

Performance comparison:

Efficiency: Partition-based verification can be more efficient than point-wise verification, especially for large networks.

Accuracy: The accuracy of robustness verification depends on the robustness metric used and the quality of the partitioning in partition-based verification.

Scalability: DNNs and CNNs can be computationally expensive to verify, especially for large networks. Partition-based verification can help improve scalability.

Researchers are working harder than ever in applied physics and engineering to improve and understand the valuable properties of nanomaterials. The past few years have seen a surge in interest in nanofluids ^[11,12]. The extreme heat conductivity of microfluids sets them apart from ordinary fluids. The enhancement in thermal conductivity of nanofluids has overtaken the restricted transfer of heat performance of conventional fluids currently in use as an extremely important phenomenon. Researchers are drawn to the exciting and crucial analysis of nanofluids because of the potential uses, for instance safer surgery, electrical gadgets, cancer therapy and nuclear reactors may boost heat transfer resources for a better cooling method, Nayak et al. ^[13]. Choi and Eastman et al. ^[14] seemed to be the initial researchers to report the dispersion of nanoparticles in a base fluid in 1995. Sheikholeslami et al. ^[15] studied the features of nanofluids, such as viscosity, conductivity, and thermal property, which have a substantial influence on heat transfer rates. There is now a large body of work on the mixed convection nanofluid flow stagnation points. Syam et al. ^[16] investigated the dynamics of Fe₃O₄–water, Cu–water, and Ag–water nanofluids in the context of steady, two-dimensional, incompressible laminar magnetohydrodynamic (MHD) boundary layer flow. Makinde et al. ^[17] explored the influence of the magnetic field and buoyancy pressure at a conduction-heated medium on the stagnation point flows. Metals, carbides, oxides, and carbon nanotubes are often discovered to include nanoparticles. In the fields of biomedicine, computer microprocessors, nuclear reactor cooling, and industry, nanofluids are broadly exploited to enhance the heat transfer ^[18,19]. Syam et al. ^[20] investigated the slip flow dynamics involving Al₂O₃ and Fe₃O₄ nanoparticles within a horizontal channel embedded with porous media. Nadeem et al. ^[21] investigated a unique kind of nanofluid; they found that the nanoparticles could have been tabular or rod-shaped.

The flow caused by disk rotation is currently of much curiosity because of its extensive range of uses in engineering and aeronautical science branches, for example, medical apparatus, thermal power-producing systems, rotating equipment, computer-storing tools, gas turbine blades, crystal growth methods, etc. Von Karman ^[22] pioneered the flow caused by a revolving disk. His well-known work has been explored by many scholars throughout the world through varied approaches. Turkyilmazoglu and Senel ^[23] investigated a combined impact of heat and mass transfer phenomena in the rotating flow of a viscous fluid over a permeable disk. They analyzed the numerical outcomes of the flow problem. Rashidi et al. ^[24] investigated the MHD viscous fluid flow via a spinning disk with various parameters. In this research, they also looked at how entropy creation affected the flow of the problem under consideration. Turkyilmazoglu ^[25] presented on the flow of a revolving disk while taking nanoparticle impacts into consideration. Hatami et al. ^[26] investigated the laminar, incompressible viscous nanoliquid flow produced by revolving and compressing disks. They performed the study using the smallest squares approach. Mustafa et al. ^[27] have studied the flow caused by a linear extending disk in the existence of nanoparticles. They came to the conclusion that a major factor in lowering the thickness of the boundary coating is the uniform stretching of the disk. Sheikholeslami et al. ^[28] reported the numerical solutions for the nanofluid flow issue affected by an angled spinning disk.

Currently, increased heat transmission rates are required by industrial production processes. The appropriate heat transfer rates required by industry are not being fulfilled by the traditional heat transfer technologies. The main focus of contemporary scientists working on this topic is the development of methods for improving heat transmission. Enhancement methods may be broadly divided into two groups: Active as well as passive procedures. While active methods need external forces like magnetic and electric fields, passive techniques just require certain fluid extracts, thermal packing, geometries, etc. Enhancing the thermal execution of common liquids, including water, oil, and ethylene-glycol combinations is the main goal of fluid additives ^[29]. Numerous researchers have documented how solid-liquid mixtures significantly increase heat transmission. Various problems have been identified as the reason why this combination was not suitable for improving thermal performance, including clogging, abrasion, and increased pressure loss.

There has been a lot of importance in the analysis of MHD boundary coating flow and heat transfer in recent years because of its applications in engineering and technology. Initially, astrophysical and geophysical problems were resolved with MHDs. Among its many applications were in silk drift separation, drug delivery, biomedicine, optical modulators, magnetic cell separation, optical grids, and optical switches, and nonlinear optical materials. MHDs are concerned with the flow of an electrically behaving fluid in a magnetic field, which has the power to regulate system flow and heat transfer. In 1974 the MHD boundary coating flow of an electrically affecting fluid in the existence of a constant transverse magnetic field was investigated by Pavlov et al. ^[30]. Gupta and Chakrabarti et al. ^[31] built on this work by observing MHD flow and heat transfer in 1979. When a magnetic field was introduced, Hayat et al. ^[32] observed the MHD flow of Sisko nanofluid and hypothesized that the bidirectional stretching of the surface was the cause of the flow production. Heysiattalab et al. ^[33] inspected the anisotropic action of magnetic nanofluids throughout film-wise condensing across a vertical plate with a constant variable-directional magnetic field. Hayat et al. ^[34] investigate the MHD three-dimensional nanofluid flow via a nonlinear stretching surface that had been convectively heated. Hayat et al. ^[35] investigated the stratified flow of a thixotropic nanofluid in the presence of magnetic field effects. Malvandi et al. ^[36] investigated the thermal properties of hydromagnetic nanofluid flow in a vertical micro annular tube. Hayat et al. ^[37] have presented an analytical approach for the three-dimensional Oldroyd-B nanofluid flow with heat generation and preoccupation. Examining the mean hydrodynamic drag (MHD) of a viscous nanoliquid as a result of a spinning disk is the main objective of this study. Interactions between nanoparticles cause Brownian motion and thermophoretic phenomena. We applied the velocity slip scenario instead of the no-slip requirement. Very often, velocity slip may occur between the moving surface and the fluid when the liquid in question is particulate, as is the case with rarefied gas, suspensions, mixes, and so on. In these cases, the slip condition, which is defined by Navier's equation, determines the proper boundary condition.

Investigator employ a pattern of numerical and semi-numerical methods to solve problems when getting a precise analytical resolve may be challenging. Methodologies include spectral relaxation ^[38], the HAM method ^[39], the Galerkin finite element method ^[40], the Keller box method ^[41], and many others. Evolutionary approximation approaches have been employed in stochastic numerical computing algorithms linked to neural networks to identify the solutions/outcomes of differential equations for both linear and nonlinear cases, showing separate evaluations for different parameters. Thermodynamics ^[42], magnetohydrodynamics ^[43,44], COVID-19 models ^[45,46], nanotechnology ^[47], mosquito dispersal models ^[48], flow model of micro polar nanofluid flow ^[49], Emden Fowler systems ^[50,51], nonlinear corneal shape models ^[52] and atomic physics ^[53] are among the methodologies that are used.

By analyzing the effects of all alterations or physical measurements on profiles of velocity, concentration, and temperature, and exploiting numerical and graphical studies, these compelling characteristics encourage researchers to exploit a reputable and precise AI algorithm-based numerical computational framework for numerical investigation of the MHD nanofluid mathematical model. MATLAB and Mathematica both were operated to improve numerical performance.

The innovation contributions of the present study for backpropagation networks for the magnetohydrodynamic nanofluid flow due to the rotating disk (MHD-VNRD) model are highlighted as follows.

➢ The numerical computation has been designed through the neural networks with the backpropogation Levenberg-Marquardt scheme (BPLMT-NN) for the comparative study of the impact of magnetohydrodynamic nanofluid flow over a rotating disk (MHD-VNRD) with slip effects.

➢ The governing mathematical model of the impact of magnetohydrodynamic nanofluid flow over a rotating disk (MHD-VNRD) with slip effects represented with nonlinear PDEs is reduced to a nonlinear system of ODEs by the competency of similarity adjustments.

➢ Reference data of designed networks is constructed effectively for variants of MHD-VNRD demonstrating the scenarios for the velocity slip parameter, parameter of Brownian motion, magnetic field parameter, Prandtle number, thermophoresis parameter, and Lewis number by applying the RK4 method.

➢ Mathematica software is used to compute the dataset for designed BPLMT-NN for the variation of the velocity slip parameter, the parameter of Brownian motion, magnetic field parameter, Prandtle number, thermophoresis parameter, and Lewis number.

➢ MATLAB software is used to interpret the solution and the absolute error analysis plots of MHD-VNRD.

➢ The training, testing, and validation-based process block structure of BPLMT-NN exploited to calculate the approximate solutions of MHD-VNRD and comparative study validate the consistent accuracy.

➢ The worthy performance of the designed network was additionally established by the learning curve on MSE-based fitness, histograms, and regression metrics.

The study has been organized as follows:

The purpose and implications of the MHD-VNRD paradigm are discussed in Section 2. The solution methodology and the impact of the suggested BPLMT-NN on many MHD-VNRD alternatives are covered in Section 3. The findings evaluation is covered in Section 4, and some last thoughts and suggestions for future study are contained in Section 5.

2.   Mathematical interpretation and evaluation of flow

We study the steady-state MHD boundary coating flow caused by a rotating disk under the condition of velocity slip, resulting in an incompressible viscous nanofluid with $z = 0$, and the disk rotating with an angular velocity Ω, which is constant. When a homogeneous magnetic field of intensity ${B}_{0}$ is employed in a certain direction, a viscous fluid conducts electricity (see Figure 1). Also, the electric field and Hall current are not taken into consideration. A low magnetic Reynolds number has no bearing on the generated magnetic field. Aspects of heat transfer and mass are studied when Brownian motion and thermophoresis are present. The directions of escalating $(r, \phi, z)$ correspond to the velocity components $(u, v, w)$. According to ^[25,28], the following boundary coating expressions control the current flow state:

The related boundary conditions are

Where $\mu$ represents the dynamic viscosity, ${\rho }_{f}$ denotes the density of base fluid, $\nu ( = \frac{\mu }{{\rho }_{f}})$ the kinematic viscosity, $\sigma$ is the electrical conductivity, $\alpha = \frac{k}{{{(\rho }_{c})}_{f}}$ denotes the thermal diffusivity of fluid, $T$ is the temperature, $k$ represents the thermal conductivity of fluid, ${{(\rho }_{c})}_{p}$ is the effective heat capacity of nanoparticles, ${{(\rho }_{c})}_{f}$ is the heat capacity of fluid, ${D}_{B}$ is the coefficient of the Brownian diffusion, ${D}_{T}$ denotes the coefficient of the thermophoretic diffusion, $C$ is the concentration, $L$ is the velocity slip constant, ${C}_{w}$ shows the concentration at the surface ${C}_{\infty }$ represents the concentration far from the surface ${T}_{w}$ shows the temperature at the surface, and ${T}_{\infty }$ represents the temperature far from the surface.

The subsequent dimensionless variables are now introduced:^[54]

Now Eq (1) is fulfilled and Eqs (2)–(7) become to the subsequent structures:

The converted boundary conditions are defined as:

Here $\gamma$ denotes the parameter of velocity slip, $M$ is the magnetic parameter, the Prandtl number is represented by $Pr$, $Nt$ is the thermophoresis parameter, $Le$ is the Lewis number, and $Nb$ is the parameter of the Brownian motion. All the parameters are expressed as follows:

The non-dimensional kind of skin friction coefficients and local Nusselt and Sherwood values can be found as

3.   Methodology and evaluation

In Figure 2, the proposed BPLMT-NN is displayed within a neural network. Using 'nftool', a method for fitting a neural network in MATLAB, the suggested BPLMT-NN is achieved. Levenberg-Marquardt backpropagation is utilized to find the weight of neural networks.

The suggested BPLMT-NN approach is shown in Figure 3, while Figure 4 shows a single flow diagram that illustrates the whole study process. Using 'nftool', six different MHD-VNRD model variants are analyzed; BPLMT-NN provides the correct solution.

Figures 5–8 show how the BPLMT-NN affects the MHD-VNRD model for Scenarios 1–6. The error histogram, fitting, and MSE performance diagrams for each of the six Case 4 scenarios $M, \gamma, Le, Nb, Nt, \mathrm{a}\mathrm{n}\mathrm{d}\;Pr$ are displayed in Figures 5(ⅰ) and 5(ⅱ). Figures 6(ⅰ) and 6(ⅱ) show the state transitions and regression analysis for all of the four MHD-VNRD model cases. Table 1 display illustration of four separate cases and all six MHD-VNRD model scenarios, while Tables 2 and 3 display the convergence achieved parameters for Case 4 of Scenarios 1–6 in the form of executed period, performance duration, backpropagation, MSE, execution, and sequential complicated gauges.

The MSE evolutions, training, and validation convergences for Scenarios 1–6 of the MHD-VNRD model are shown in Subfigures 5(ⅰ) and 5(ⅱ) with regard to testing procedures. The suggested approach performs more accurately and efficiently when the MSE value is lower. With MSE close to $1.24\times {10}^{-10}, 2.61\times {10}^{-10}, 1.93\times {10}^{-09}, 2.58\times {10}^{-10}, 7.09\times {10}^{-10}, \mathrm{a}\mathrm{n}\mathrm{d}\;3.16\times {10}^{-11},$ the greatest network performance is achieved.

The produced results of six distinct scenarios with a step size of 0.01 and inputs ranging from 0 to 5 are examined in Figures 5(ⅰ)–5(ⅱ) to illustrate the effectiveness of the MHD-VNRD model. By connecting the obtained results to explicit Runge-Kutta technique reference numerical results and related results, the error dynamics plot is validated. The error histograms in Figures 5(ⅰ)–5(ⅱ) are just one part of the study. The MHD-VNRD model's error dynamics and results for Case 4 of different situations are also looked at for all input points. It was found that the suggested BPLMT-NN has a maximum error of less than $1.85\times {10}^{-10},$ $5.08\times {10}^{-10},$ $2.00\times {10}^{-09},$ $2.76\times {10}^{-10},$ $8.53\times {10}^{-10}$ and $1.66\times {10}^{-05}$ for Case 4 of the figures 1 to 6 scenarios of the design model. For point 4 of the Figures 1 to 6 various scenarios of MHD-VNRD, the error bin average value relative to the zero line has error near $-3.3\times {10}^{-06}, 1.01\times {10}^{-05}, -7.6\times {10}^{-06}, 4.53\times {10}^{-06}, -7.42\times {10}^{-06},$ and $-2.2\times {10}^{-06}$. In relation to the values of the Levenberg-Marquardt gradient and step size of Mu, the following values are strongly related: $\left[1.00\times {10}^{-08}, 1.00\times {10}^{-08}, 1.00\times {10}^{-08}, 1.00\times {10}^{-08}, 1.00\times {10}^{-08}, \mathrm{a}\mathrm{n}\mathrm{d}\;1.00\times {10}^{-085}\right]$ and $[9.97\times {10}^{-08}, 9.96\times {10}^{-08}, 9.99\times {10}^{-08},$ $9.99\times {10}^{-08}, 9.96\times {10}^{-08}, \mathrm{a}\mathrm{n}\mathrm{d}\;9.97\times {10}^{-08}]$ are displayed in Subfigures 6(ⅰ) and 6(ⅱ). BPLMT-NN is competent, dependable, and convergent, as demonstrated by the outcomes and graphical depictions for the MHD-VNRD model in Case 4. Corelation analyses are frequently used to classify data in regression analysis. The correlation R values exhibit a consistent proximity to one, indicating optimality for testing, training, and validation in certain modeling scenarios. This indicates the MHD-VNRD model resolution of the BPLMT-NN. Furthermore, as in 4 of the several MHD-VNRD model Scenarios 1–6, the corresponding numerical values in Tables 2 and 3 show that the MSE performance for the recommended BPLMT-NN is close to ${10}^{-11}$. Tables 2 and 3's numerical results demonstrate how well the BPLMT-NN executes the MHD-VNRD model.

For the temperature profile $\theta \left(\eta \right)$), concentration ϕ(η), and radial and tangential velocities f'(η) and g(η), the BPLMT-NN effects are confirmed. They are shown in Figures 7(ⅰ) and 7(ⅱ), respectively, for each of the six MHD-VNRD framework scenarios. Subfigures 7(ⅰ)(a,b,c) illustrate the impact of velocity profiles f'(η), g(η), and temperature profiles θ(η) on the fluctuation of Lewis number Le, magnetic parameter M, and velocity slip parameter γ for the MHD-VNRD model in Scenarios 1–3. The impact of magnetic parameter M on the velocity profile f'(η) is seen in Figure 7(ⅰ)(a). For increasing values of the magnetic parameter, there is a reduction in both the velocity profile and the thickness of the momentum boundary layer. Higher values of the velocity slip parameter γ result in a lower velocity profile g(η) and a thinner momentum boundary layer, as seen in Figure 7(ⅰ)(b). This shows how a higher Lewis number Le results in a thinner concentration layer and a lower concentration profile ϕ(η). In light of this, the temperature profile $\theta \left(\eta \right)$ and concentration ϕ(η) for Scenarios 4–6 in the MHD-VNRD model are represented by different values in Subfigures 7(ⅱ)(a,b,c), for the Brownian motion parameter Nb, thermophoresis parameter Nt, and Prandtl number Pr, respectively. The effect of the parameter Nb for Brownian motion on the temperature profile θ(η) is seen in Figure 7(ⅱ)(a). When the bigger values of the Brownian motion parameter are taken into consideration, it is found that an improvement occurs in the temperature profile. Greater temperature and thermal boundary layer thickness are produced by larger Brownian motion parameters, which also have lesser viscous forces and greater Brownian diffusion coefficients. The impact of the thermophoresis parameter Nt on the concentration profile ϕ(η) is seen in Figure 7(ⅱ)(b). Concentration profile and matching concentration layer thickness both grow in relation to the values of the thermophoresis parameters in this case. To make matters worse, increasing the thermophoretic force makes things worse by physically trying to stimulate the migration of nanoparticles in the opposite direction as the temperature gradient, which means that the concentration gradient is larger and the concentration profiles are less uniform. As seen in Figure 7(ⅱ)(c), a decreased temperature profile θ(η) is caused by the Prandtl number Pr. It demonstrates that a decreasing temperature profile θ(η) and decreasing thermal boundary layer thickness are caused by a growing Prandtl number Pr. The Prandtl number is physically dependent on the diffusivity of heat. A lower thermal diffusivity is correlated with larger Prandtl numbers. The temperature profile and corresponding thickness of the thermal boundary layer decrease as a result of this reduced thermal diffusivity.

4.   Conclusions

The magnetohydrodynamic investigation of the boundary layer flow of a viscous nanofluid due to the rotation of a disk under velocity slip conditions is carried out. Properties of heat transfer and mass are investigated when thermophoresis and Brownian motion are present. The backpropagation Levenberg-Marquardt approach combined with neural networks (BPLMT-NN) yields numerical solutions. The important facts of the current study are stated as follow:

Higher magnetic parameter values decrease velocity distributions, while temperature and concentration distributions exhibit the reverse tendency.

The backpropagation Levenberg-Marquardt technique with neural networks (BPLMT-NN) has been used to examine the solution of a mathematical model that displays the slip effect with modifications of certain situations or scenarios.

By using relevant connected variables, a mathematical flow could be translated into a structure of nonlinear ODEs, which can be represented by PDEs.

The explicit Runge-Kutta technique has been utilized to produce the dataset for the (MHD-VNRD) model.

The deviations from several physical data, including the chemical reaction factor, velocity slip parameter, Prandtl number, magnetic field, thermophoresis, and Brownian motion and parameters, were included.

Both of the velocity's components decrease as the velocity slip parameter increases.

The distributions of concentration and temperature improve as the thermophoresis parameter increases.

A number of variants of the MHD-VNRD reference data set are altered utilizing the BPLMT-NN for training, testing, and validation, using 80%, 10%, and 10% of the data set, respectively.

The scheme's brilliance receives a ${10}^{-9}\;\mathrm{t}\mathrm{o}\;{10}^{-11}$ grade for both reference and suggested results.

The functionality that has been explained is bolstered by both numerical and graphical depictions of regression dynamics, mean square errors, and convergence error-histogram graphs.

Future applications of the BPLMT-NN concept and its most recent improved versions might be in fluid dynamics problems ^[55,56] and a variety of computer network propagation models ^[57,58].

Author contributions

Yousef Jawarneh: Conceptualization, writing-review and editing, project administration; Samia Noor: Visualization, writing-review and editing, data curation, funding; Ajed Akbar: Data curation, project administration; Rafaqat Ali Khan: Resources, validation; Ahmad Shafee: Software, investigation, resources. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Funding

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. KFU251494].

Conflict of interest

The authors declare that they have no conflicts of interest.