Computational Intelligence Approach for Optimising MHD Casson Ternary Hybrid Nanofluid over the Shrinking Sheet with the Effects of Radiation

Abstract

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.

1. Introduction

The fluid whose viscosity depends upon the shear-thinning/thickening properties of fluid is known as Non-Newtonian. Based on viscosity dependence characteristics different fluids models have been developed namely Casson fluid, Maxwell fluid and Walter B’fluid models etc. Non-Newtonian fluids have many engineering applications i.e., petroleum industry, plastic and polymer industry, printing technology, food preservation and optical fibre industry. Transport phenomena of several fluids models (Non-Newtonian) were examined by numerous scientists [1,2,3,4,5]. Casson fluid model was named after the name of researcher Casson (for details see Ref. [6]) who developed the non-Newtonian fluid model whose viscosity is high at $\beta =0$ and low when $\beta =\infty$. This model depicts the liquid properties that occur when shear stress exceeds yield stress, illustrative of shear-thinning occurrence in nature. The unsteady Casson fluid was examined by Mythili and Sivaraj [7] and they concluded that fluid velocity increases by increasing $\beta$, which is a Casson fluid parameter. Mukhopadhyay et. al. [8] scrutinized the flow behaviour of non-Newtonian fluid over the non-porous wedge and concluded that $N{u}_x$ shows an increment with an increasing Casson parameter and thermal radiation effect. Mukhopadhyay and Mandal [9] investigated boundary layer flow over a porous wedge and the effects of $\beta$ are discussed in flow situations. They observed that when values of $\beta$ are increased then $C{f}_x$ is increased.

Nanofluids are fluids that are enhanced with nanoparticles. The nanoparticles, which are typically made of metals or metal oxides are dispersed in fluids typically a liquid such as water, to form a suspension. The addition of nanoparticles to a fluid has been shown to improve its thermal conductivity ($Rd$), making it a promising material for use in heat transport phenomena and applications such as cooling in electronics and automotive systems. Nanofluids have also been studied for their potential use in other applications, including biomedical imaging and drug delivery. However, more research is required to fully understand the characteristics and transport properties of nanofluids and their potential ability for real-world applications. Additionally, there are apprehensions about the safety and environmental impact of nanoparticles, which must be carefully considered before nanofluids are widely used. Nanofluids have novel properties and characteristics in the science and engineering industry i.e., pharmaceutical and biomedicine industry, electronics devices, domestic used appliances like refrigerators, and engineering devices in power and chemical engineering. Nanofluids are categorized as solid-liquid combinations that consist of nanoparticles (1–100 nm) as well as a carrier medium (base fluid). Nanofluids were named after the name of Choi [10]. The numerical investigation of boundary layer flow over an expanding wedge had wall velocity $u_w\left(x\right)$ and free stream (abient) velocity $u_e\left(x\right)$ was discussed [11]. An investigation of heat transfer and unsteady boundary layer flow across porous and exponential shrinking sheets filled with $cu$-water nanofluid was scrutinized [12]. Ellahi et al. [13], scrutinized a 2-D and incompressible flow of nanofluid with heat transfer at the stagnation point having mixed convection. The 2-D flow of viscoelastic fluid over an extending surface with an assumption of the ferromagnetic effect produced by magnetic dipole was explored by Hassan [14].

A hybrid nanofluid is categorized as a nanofluid that consists of various nanoparticles suspended in a base fluid. These nanoparticles might be metallic, nonmetallic, or a mix of the two. The hybrid aspect of the nanofluid results from synergistic effects caused by the interaction of several types of nanoparticles, which can result in improved properties when compared to individual-component nanofluids. The numerical investigation of a hybrid nanofluid was investigated over an exponentially shrinking sheet with an assumption of suction/injection parameters [15]. Zeeshan et al. [16] scrutinized the outcome of $Rd$ on $A{l}_2{O}_3$-EG nanofluid through a porous wavy channel, discussing the consequence of various pertained parameters on $C{f}_x$ and $N{u}_x$ at the top and bottom of the channel. Bhatti et al. [17] presented a paper on MHD Williamson nanofluid flow of gyrotactic microorganisms in a porous medium, using DTM-Pade to solve mathematical models and obtain numerical results of various parameters on transport phenomena such as Nusselt number. Kolsi et al. [18] investigated the thermal properties of small viscoelastic particles moving in a curved channel. It appears that the peristaltic motion in this situation is being created to assist or direct the movement of the minute viscoelastic particles. A hybrid nanofluid ($F{e}_3{O}_4-$MWCNT/water) experiences mixed convection heat transfer, flow behaviour, and entropy creation as a result of the lid-driven top wall’s effect and side walls’ waviness map. This process is examined by Maneengam et al. [19]. Other interesting conclusions regarding such fluids can be found in [20,21,22,23].

Due to the intricate nature of preparation and the relatively unstable characteristics of ternary hybrid nanofluids, limited research has been conducted on their thermophysical properties. Sahoo et al. [24] scrutinized the influence of temperature and volume fraction on $A{l}_2{O}_3-CuO-Ti{O}_2/W$ ternary hybrid nanofluid, revealing a substantial increase in viscosity of ternary hybrid nanofluid in comparison to corresponding binary hybrid nanofluids. Subsequently, a similar investigation was carried out on $A{l}_2{O}_3-SiC-Ti{O}_2/W$ ternary hybrid nanofluid [25]. The significant properties of nanofluid, hybrid nanofluid, ternary hybrid nanofluid, nanoparticles and base fluid are given in

Table 1. Thermo-physical properties of nanofluid, hybrid nanofluid, and, ternary hybrid nanofluid [24,25].

Properties	Hybrid Nanofluid	Ternary Hybrid Nanofluid
$\mu$	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_1 \right)}^{\left(2.5\right)}{\left(1-{\varphi }_2 \right)}^{\left(2.5\right)}}$	${\mu }_{thnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_1 \right)}^{\left(2.5\right)}{\left(1-{\varphi }_2 \right)}^{\left(2.5\right)}{\left(1-{\varphi }_3 \right)}^{\left(2.5\right)}}$
$\rho$	${\rho }_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_f+{\varphi }_1{\rho }_{p1}\right]+{\varphi }_2{\rho }_{p2}$	${\rho }_{thnf}=\left(1-{\varphi }_3\right)\left[\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_f+{\varphi }_1{\rho }_{p1}\right]+{\varphi }_2{\rho }_{p2}\right]+{\varphi }_3{\rho }_{p3}$
$C_p$	${\left(\rho C_p\right)}_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho C_p\right)}_f+{\varphi }_1{\left(\rho C_p\right)}_{p1}\right]+{\varphi }_2{\left(\rho C_p\right)}_{p2}$	${\left(\rho C_p\right)}_{thnf}=\left(1-{\varphi }_3\right)\left[\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho C_p\right)}_f+{\varphi }_1{\left(\rho C_p\right)}_{p1}\right]+{\varphi }_2{\left(\rho C_p\right)}_{p2}\right]+{\varphi }_3{\left(\rho C_p\right)}_{p3}$
$k$	$\frac{k_{hnf}}{k_{nf}}=\frac{k_{p2}+2k_{nf}-2{\varphi }_2\left(k_{nf}-k_{p2}\right)}{k_{p2}+2k_{nf}+{\varphi }_2\left(k_{nf}-k_{p2}\right)}$ Where $\frac{k_{nf}}{k_f}=\frac{k_{p1}+2k_f-2{\varphi }_1\left(k_f-k_{p1}\right)}{k_{p1}+2k_f+{\varphi }_1\left(k_f-k_{p1}\right)}$	$\frac{k_{thnf}}{k_{hnf}}=\frac{k_{p3}+2k_{hnf}-2{\varphi }_3\left(k_{hnf}-k_{p3}\right)}{k_{p3}+2k_{hnf}+{\varphi }_3\left(k_{hnf}-k_{p3}\right)}$ Where $\frac{k_{hnf}}{k_{nf}}=\frac{k_{p2}+2k_{nf}-2{\varphi }_2\left(k_{nf}-k_{p2}\right)}{k_{p2}+2k_{nf}+{\varphi }_2\left(k_{nf}-k_{p2}\right)}$
$\sigma$	$\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_{p2}+2{\sigma }_{nf}-2{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{p2}\right)}{{\sigma }_{p2}+2{\sigma }_{nf}+{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{p2}\right)}$ Where $\frac{{\sigma }_{nf}}{{\sigma }_f}=\frac{{\sigma }_{p1}+2{\sigma }_f-2{\varphi }_1\left({\sigma }_f-{\sigma }_{p1}\right)}{{\sigma }_{p1}+2{\sigma }_f+{\varphi }_1\left({\sigma }_f-{\sigma }_{p1}\right)}$	$\frac{{\sigma }_{thnf}}{{\sigma }_{hnf}}=\frac{{\sigma }_{p3}+2{\sigma }_{hnf}-2{\varphi }_3\left({\sigma }_{hnf}-{\sigma }_{p3}\right)}{{\sigma }_{p3}+2{\sigma }_{hnf}+{\varphi }_3\left({\sigma }_{hnf}-{\sigma }_{p3}\right)}$ Where $\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_{p2}+2{\sigma }_{nf}-2{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{p2}\right)}{{\sigma }_{p2}+2{\sigma }_{nf}+{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{p2}\right)}$

and

Table 2. Physical properties of nanoparticles and base fluid [24,25].

Properties	$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$	$\mathit{C}\mathit{u}$	$\mathit{T}\mathit{i}{\mathit{O}}_2$	Water
$C_p\left(\frac{J}{KgK}\right)$	686.2	385	711	4179
$\rho \left(\frac{kg}{m^3}\right)$	6310	8933	4250	997.1
$k\left(\frac{W}{mK}\right)$	32.9	400	8.953	0.613
$\sigma \left(\frac{W}{m^2}\right)$	36,900,000	59,600,000	2,380,000	0.05
$Pr$	-	-	-	6.2

. The existing viscosity prediction model deviated significantly from experimental data, prompting the introduction of a new correlation based on $A{l}_2{O}_3-SiC-Ti{O}_2/W$ ternary hybrid nanofluid. Mousavi et al. [26] investigate the outcomes of three different nanoparticles like $CuO-MgO-Ti{O}_2/W$ on the transport properties of ternary hybrid nanofluid. Cakmak et al. [27] deliberated the outcomes of nanoparticles on flow stability and thermal conductivity ($Rd$) of $rGO-F{e}_3{O}_4-Ti{O}_2/EG$ ternary hybrid nanofluids, revealing that thermal conductivity ($Rd$) enhancement significantly increases with mass concentration and temperature.

Magnetohydrodynamic (MHD) boundary layer research has fascinated the attention of various academics in recent decades owing to its wide range of applicability in engineering and technology. The magnetic effect of nanofluids refers to the behaviour of the fluid in response to a magnetic field. The metallic particles in a fluid can alter its magnetic properties, leading to various effects i.e., magnetic susceptibility and magnetic relaxation. For example, magnetic refrigeration, MHD pumps, a promising technology for energy-efficient cooling, flow metres and several electrically conducting fluid. The numerical study of MHD effect on the flow situations of the viscous fluid over a non-isothermal wedge was discussed by Yih [28] and Chamkha [29]. For more applications of the MHD in heat and mass transfer see Ref. [30,31,32,33,34,35].

The researcher uses a variety of numerical approaches to handle problems when an exact analytical answer cannot be found. The homotopy perturbation technique [36], Keller-Box [37], spectral relaxation method [38], finite element analysis [39], and many more [40,41]. Numerous numerical approaches have been used to solve problems in the aforementioned studies on different nanofluid systems, but due to their value and effectiveness, AI-based numerical computing strategies are important for obtaining an approximate solution to the boundary value problem of nanofluid. The broad category of artificial intelligence includes deep learning, which contains neural networks. Artificial neural networks (ANNs) are powerful mathematical tools applied for numerous purposes, like data classification, self-driving cars, and stock market predictions. Artificial neural networks were created with motivation from the human nervous system. The neural network is a computer processing information paradigm for raw data. As in biological neurons in the human system, the artificial neurons in neural networks, in which mathematical functions are developed to build correlation among physical parameters. An estimated 10 billion neurons, each coupled with 10,000 others on average, make up the human brain. These neurons compile the raw data and give valuable results after performing suitable mathematical operations. ANNs have gained importance due to their applications in fluid dynamics and many researchers implement these approaches to solve the boundary value problem. ANNs have found a wide range of applications in fluid flow modelling due to their ability to learn and generalize complex relationships between input and output data. ANNs are a type of machine learning algorithm that can be trained to predict or classify data based on input features. In fluid flow, ANNs have been used to model a wide range of phenomena, including turbulent flows, heat transfer, and multiphase flows. ANNs can also be used to optimize the performance of fluid flow systems. For example, ANNs can be applied to optimize the shape of a nozzle or valve to minimize pressure drop or maximize flow rate. ANNs can also be used to control fluid flow systems, such as in active flow control or flow rate regulation. Shoaib et al. [42], deliberated the impact of thermal energy on magnetohydrodynamic processes in terms of heat source ($Q$) and thermal radiation ($Rd$) on Casson fluid flow model (MHD-CFM) over a nonlinear slanted extending surface. They used the Levenberg Marquardt methodology, which is a trained neural network. Artificial neural networks were applied to the nanofluid having aluminium oxide ($A{l}_2{O}_3$) and silver (Ag) as nanoparticles and water as base fluid over an extending sheet. Khan [43], examined the comparison of the Levenberg-Marquardt scheme (LMS) and Bayesian regularization backpropagation (BRBP) for partial differential equations. The ANNs technique is widely adopted by different researchers, for more details see Ref. [44,45,46,47,48]. To the best of the authors’ knowledge, there is no one working on the ternary hybrid nanofluid with ANNs simulation.

2. Background

Consider 2-D steady, and incompressible boundary layer flow of Casson ternary hybrid nanofluid over a porous lessening sheet with uniform surface temperature. The ternary hybrid nanofluid model is utilized to model flow equations with an assumption of magnetic ($M$) and thermal radiation ($Rd$) effect on a contracting sheet. There are three different nanoparticles are used. The red nanoparticles represent Aluminium oxide ($A{l}_2{O}_3$), green particles show Titanium oxide ($Ti{O}_2$) and yellow particles show the cupper ($Cu$). $u_w$ represents wall velocity and $u_e$ is ambient velocity.

Figure 1 depicts the schematic diagram of a contracting sheet. $T_w$ are the temperature at the surface of a sheet where $T_{\infty }$ is ambient temperature. Additionally, a constant magnitude of a magnetic field is implemented in $y$-aixs. The under-consideration flow equations of Casson nanofluid [49].

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=u_{\infty }\frac{d{u}_{\infty }}{dx}+\left(\frac{{\mu }_{thnf}}{{\rho }_{thnf}}\right)\left(1+\frac{1}{\beta }\right)\nu \frac{{\partial }^{2}u}{\partial y^2}+\frac{{\sigma }_{thnf}}{{\rho }_{thnf}}{B}^2\left(u_{\infty }-u\right),$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\left(\frac{1}{{\left(\rho cp\right)}_{thnf}}\left(k_{thnf}+\frac{16\sigma {T_{\infty }}^3}{3K}\right)\frac{{\partial }^{2}T}{\partial y^2}\right).$$

(3)

Boundary conditions of governing flow problems are;

$$\begin{gathered} u=u_w\left(x\right),v=v_w\left(x\right),T=T_w\mathrm{at}y=0, \\ u\to u_e\left(x\right),T\to T_{\infty }\mathrm{as}y\to \infty . \end{gathered}$$

(4)

where $u$ and $v$ are velocity components along $x$ and $y$-direction. The temperature of the fluid is represented by $T$. Here, $\nu$ is kinematics viscosity, $B\left(x\right)=B_0{x}^{\left(m-1\right)/2}$, $B_0$ is constant (see Kudenatti et al. [50]), and $q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial y}$ (for detail see reference [51]). The flow Equations (1)–(3) and boundary condition (4) are PDEs, to convert these equations into ODEs we exhibit an efficient similarity transformation as follows;

$$\begin{gathered} \mathsf{\psi }\left(x,y\right)=f\left(\eta \right)\sqrt{\frac{2\nu }{a\left(n+1\right)}{x}^{n-1}},\eta =y\sqrt{\frac{a\left(n+1\right)}{2\nu }{x}^{n-1}y},\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}, u_e=U_e{x}^n, \\ {u}_w=U_w{x}^n. \end{gathered}$$

(5)

The dimensionless variables/ transformation is presented in Equation (5). These dimensionless variables satisfied the continuity equation identically. However, the Equations (2) and (3) and boundary conditions Equation (4) are transformed into a system of ODEs and dimensionless boundary conditions by applying similarity variables and velocity components $u$ and $v$ as follows;

$$\left(\frac{{\mu }_{thnf}/{\mu }_f}{{\rho }_{thnf}/{\rho }_f}\right)\left(1+\frac{1}{\beta }\right)f^{‴}\left(\eta \right)+f\left(\eta \right)f^{″}\left(\eta \right)+\frac{2n}{n+1}-\frac{2n}{n+1}{f}^{{\prime }^2}\left(\eta \right)-\frac{{\sigma }_{thnf}/{\sigma }_f}{{\rho }_{thnf}/{\rho }_f}{M}^2{f}^{\prime }\left(\eta \right)=0,$$

(6)

$$\frac{1}{Pr}\frac{1}{\left(\frac{{\left(\rho cp\right)}_{thnf}}{{\left(\rho cp\right)}_f}\right)}\left(\frac{k_{thnf}}{k_f}+\frac{4}{3}Rd\right){\theta }^{″}\left(\eta \right)+f\left(\eta \right){\theta }^{\prime }\left(\eta \right)=0.$$

(7)

The dimensionless boundary conditions are;

$$\begin{gathered} f\left(\eta \right)=S,f^{\prime }\left(\eta \right)=\lambda ,\theta \left(\eta \right)=1,\mathrm{at}\eta =0, \\ {f}^{\prime }\left(\eta \right)\to 1,\theta \left(\eta \right)\to 0,\mathrm{at}\eta \to \infty . \end{gathered}$$

(8)

Here prime denotes differentiation w.r.t $\eta$, $\lambda =\raisebox{1ex}{$U_w$}\!\left/ \!\raisebox{-1ex}{$U_e$}\right.$ are the elongating and lessening factors with $\lambda >0$ for extending case and $\lambda <0$ for the lessening case and $S$ is a suction and injection parameters with $S<0$ for injection and $S>0$ for suction. The Prandtl number is defined as $Pr=\frac{{\left(\mu cp\right)}_f}{k_f}$. The flow response outputs are defined as follows;

$$C{f}_x=\frac{{\tau }_w}{{\rho }_f{U}_e^2},N{u}_x=-\frac{x{q}_w}{k_f\left(T_w-T_{\infty }\right)}.$$

(9)

The stress tensor ${\tau }_w$ and heat flux $q_w$ are given as follows;

$${\tau }_w={\mu }_{thnf}\left(1+\frac{1}{\beta }\right){\left(\frac{\partial u}{\partial y}\right)}_{y=0},q_w=-k_{thnf}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}.$$

(10)

Now using Equation (5) and Equations (9) and (10), we obtained the following dimensionless physical output flow response outcomes;

$$\begin{array}{c}R{e}_x^{1/2}C{f}_x=\left(1+\frac{1}{\beta }\right)\frac{{\mu }_{thnf}}{{\mu }_f}{f}^{″}\left(0\right),R{e}_x^{-1/2}N{u}_x=-\frac{k_{thnf}}{k_f}{\theta }^{\prime }\left(0\right)\end{array}$$

(11)

3. Research Method

The authors divide the solution methodology into two steps. Firstly, we conduct the numerical data sheets for LMS-BPNN by solving systems ODEs Equations (6) and (7) with boundary condition (8) using MATLAB built-in function bvp4c with the variation of Casson fluid parameters ($\beta$), MHD, suction/injection and concentration of nanoparticles and thermal radiation effects. Bvp4c is MATLAB built-in package that is based on the 3-stage Lobatto IIIa formulation. To find a numerical solution, we use suitable initial guesses. Later, using the MATLAB command ‘nftool’ from the artificial neural network (ANN) toolbox, the developed LMS-BPNN is put into practice.

Artificial Neural Network

An abstract computational framework based on the organizational layout of the human brain, artificial neural network (ANN) research is one of the most well-known fields of artificial intelligence (AI) study. ANN is a tool for data modelling that relies on a variety of parameters and learning techniques [52,53]. Usually, layers are used to organize neural networks. Layers consist of a variety of interconnected neurons/nodes which carry activation functions.

The quantity and quality of interneuron connections, which are described as numerical values called weights, is where an artificial neural network (ANN) stores the knowledge it learns. To calculate the output signal values for an initial testing input value, these weights are used. A mathematical approach called neural computing was influenced by the biological model. There are a lot of artificial neurons in this computing system, and there are even more connections between them. Different classes of neural network architecture can be distinguished based on the structure of these interconnections feed-forward neural network (FFNN), feed-backward neural network (FBNN) etc. The performance of the suggested ternary hybrid nanofluid model is investigated by LMS-BPNN. Feedforward neural networks are the most prevalent kind of ANNs, and they are trained using the supervised learning algorithm backpropagation. Backpropagation is a technique that iteratively updates the network’s weights depending on the gradient of the loss function relative to the weights. The difference between the actual outputs (sometimes referred to as ground truth) and the anticipated outputs for a specific set of input data is measured by the loss function. A neural network is developed by taking 10 hidden layers and 3 outputs. The suggested ternary hybrid nanofluid’s LMS-BPNN performance is validated by regression estimation, histogram research, and examination of the MSE values using the ‘nftool’ command. To run the planned LMS-BPNN, the outcome for velocity and temperature profile for inputs 0 to 5 is widely scattered, and the numerical data-sheet is divided into 80% of training, 10% of validation and 10% of testing data (for details see [42,43,48]).

4. Results and Discussion

The LMS–BPNN is established to scrutinize the outcomes for various pertained parameters of desired interest, $\beta$, $M$, $Rd$ and concentration of nanoparticles (${\varphi }_1$, ${\varphi }_2$, ${\varphi }_3$) for Casson ternary hybrid nanofluid over a porous lessening sheet. There are five scenarios each with three cases. The pertained parameters of interest are shown in

Table 3. Portrayal for scenarios of ternary hybrid nanofluids (for details see, refs. [42,43]).

Scenario	Case	Input Parameters
		$\mathit{\beta }$	$\mathit{M}$	$\mathit{S}$	${\mathit{\varphi }}_1={\mathit{\varphi }}_2={\mathit{\varphi }}_3$	$\mathit{R}\mathit{d}$
	1	1	0.1	0.05	0.1	0.1
1	2	2	0.1	0.05	0.1	0.1
	3	3	0.1	0.05	0.1	0.1
	1	1	0.1	0.05	0.1	0.1
2	2	1	0.5	0.05	0.1	0.1
	3	1	0.9	0.05	0.1	0.1
	1	1	0.1	0.05	0.1	0.1
3	2	1	0.1	0.1	0.1	0.1
	3	1	0.1	0.5	0.1	0.1
	1	1	0.1	0.05	0.1	0.1
4	2	1	0.1	0.1	0.05	0.1
	3	1	0.1	0.5	0.01	0.1
	1	1	0.1	0.05	0.1	0.1
5	2	1	0.1	0.1	0.1	0.5
	3	1	0.1	0.5	0.1	0.9

. Figure 2 and Figure 3 show the performance and error analysis with histograms for scenarios 1–5 for the proposed LMS-BPNN. However, Figure 4 shows the regression analysis for scenarios 1–5 using case 1 with LMS-BPNN.

Table 4. Relative examination through LMS-BPNN for all scenarios (for details see, Ref. [42,43]).

Scenario	Case	MSE Level			Performance	Gradient	$\mathit{M}\mathit{u}$	Epoch	Time (s)
		Training	Validation	Testing
1	1	8.3546 × 10−11	1.5071 × 10−8	1.8317 × 10−7	8.35 × 10−11	1.09 × 10−7	1.0 × 10−8	1000	13
	2	4.21789 × 10−9	1.97230 × 10−7	8.13650 × 10−8	4.22 × 10−9	1.09 × 10−7	1.0 × 10−7	960	12
	3	7.5426 × 10−11	1.58207 × 10−9	7.98870 × 10−9	7.54 × 10−10	9.97 × 10−8	1.0 × 10−7	925	8
2	1	2.76930 × 10−9	5.47180 × 10−6	8.25713 × 10−7	2.77 × 10−9	9.99 × 10−8	1.0 × 10−7	710	8
	2	2.28850 × 10−8	7.99340 × 10−8	4.81915 × 10−8	2.21 × 10−8	4.06 × 10−6	1.0 × 10−7	477	5
	3	5.98846 × 10−7	1.45918 × 10−5	2.75330 × 10−6	5.89 × 10−7	3.66 × 10−6	1.0 × 10−7	144	3
3	1	9.40917 × 10−9	5.93850 × 10−8	5.24300 × 10−8	9.01 × 10−9	9.61 × 10−6	1.0 × 10−8	332	5
	2	1.27770 × 10−9	6.68504 × 10−6	2.12780 × 10−5	1.28 × 10−9	9.88 × 10−8	1.0 × 10−9	776	8
	3	7.77401 × 10−9	3.86208 × 10−8	1.54685 × 10−8	7.74 × 10−9	1.51 × 10−6	1.0 × 10−6	257	4
4	1	7.07430 × 10−6	6.69880 × 10−6	1.20524 × 10−5	8.96 × 10−6	1.32 × 10−5	1.0 × 10−6	345	3
	2	1.9466 × 10−10	9.2217 × 10−10	3.83047 × 10−10	1.79 × 10−10	2.28 × 10−7	1.0 × 10−9	570	7
	3	2.5602 × 10−10	5.58543 × 10−9	2.43448 × 10−9	2.56 × 10−10	9.9 × 10−8	1.0 × 10−9	238	3
5	1	1.20131 × 10−8	5.26162 × 10−8	6.98500 × 10−8	1.20 × 10−8	9.97 × 10−8	1.0 × 10−8	721	10
	2	2.2964 × 10−10	1.18120 × 10−9	9.34810 × 10−10	2.30 × 10−10	9.97 × 10−8	1.0 × 10−8	910	13
	3	5.4312 × 10−10	1.13522 × 10−8	9.95970 × 10−9	5.43 × 10−10	9.88 × 10−8	1.0 × 10−8	881	11

shows convergence in terms of MSE for training, testing and validation, performance, epochs/iterations, time and $Mu$ is calculated for the performance of artificial neural networking. Figure 2a–e depicts a convergence plot of MSE for case 1 of all five scenarios of ternary hybrid nanofluid for training, testing and validation. The best curve is attained at 1000, 710, 332, 345, and 721 epochs, while ${10}^{-11}$, ${10}^{-9}$, ${10}^{-9}$, ${10}^{-6}$, and ${10}^{-8}$, respectively. The numerical values of $Mu$ and the gradient of LMS-BPNN are shown in Table 4. These values are [1.0 × 10⁻⁷, 1.0 × 10⁻⁷, 1.0 × 10⁻⁸, 1.0 × 10⁻⁶ and 1.0 × 10⁻⁸] and [1.09 × 10⁻⁷, 3.66 × 10⁻⁶, 9.61 × 10⁻⁶, 1.32 × 10⁻⁵, and 9.97 × 10⁻⁸]. These results have demonstrated the validity and effective convergence of LMS-BPNN for each scenario of ternary hybrid nanofluid. The examination of regression analysis can look into the information concerning correlation. Regression plots are shown in Figure 4a–e for scenarios 1–5 of ternary hybrid nanofluid. Regression analysis research can be used to investigate the facts concerning correlation. The value of correlation ($R$) is unity in Figure 4a–d, this shows that the model is best fitted. In terms of data analysis of the numerical data sheet, the value of correlation ($R$) is close to unity supporting the usefulness of LMS-BPNN for creating a ternary hybrid nanofluid. These values examine the goodness of fit and demonstrate the accuracy of modelling. The comparison of LMS-BPNN results with numerical data is shown in Figure 5a–f and Figure 6a–d for scenarios 1–5, and error analysis is used to further confirm the accuracy of results.

4.1. Influence on and Absolute Error Analysis $f^{\prime }\left(\eta \right)$

The results of the LMS-BPNN are examined using MATLAB software to explore the effects of changing the Casson fluid parameter ($\beta$), magnetic field parameter ($M$), and suction/injection effects ($S$) on $f^{\prime }\left(\eta \right)$ with absolute errors are shown in Figure 5. The absolute error analysis is conducted between numerical values and predicted values as discussed in [42,43]. Figure 5a depicts the effects of the Casson fluid parameter ($\beta$) on velocity profiles ($f^{\prime }\left(\eta \right)$) with an absolute error ${10}^{-8}\to {10}^{-4}$ as shown in Figure 5b. While Figure 5c shows the impacts of $M$ on $f^{\prime }\left(\eta \right)$ with absolute error ${10}^{-7}\to {10}^{-4}$ as depicted in Figure 5d. Similarly, the influence of porosity is depicted in Figure 5e and its absolute error (${10}^{-9}\to {10}^{-3}$) is shown in Figure 5f. The following consequences of pertained parameters on the velocity profile are observed.

• The velocity decreases as we increase the Casson fluid parameters. This effect is examined due to the shear thickening of ternary hybrid nanoparticles. It is concluded that Casson fluid parameter affect the flow behaviour, possibly interacting with the ternary hybrid nanoparticles and affecting the system’s overall shear thickening behaviour based on the observation that the velocity decreases as the Casson fluid parameters are increased.
• It is also investigated that with the enhancement in magnetic parameters the fluid velocity increases. When a magnetic field is present, it can cause Lorentz forces that can cause the fluid to move more quickly, which has an impact on magnetohydrodynamics (MHD). The Lorentz forces grow greater as the magnetic parameters are increased, increasing the fluid velocity.
• Porosity effects also enhance the velocity of the fluid. It is examined that when we increase the porosity the fluid velocity increases.

4.2. Influence on and Absolute Error Analysis $\theta \left(\eta \right)$

The results of the LMS-BPNN are examined using MATLAB software to explore the effects of changing the nanoparticles concentration (${\varphi }_1={\varphi }_2={\varphi }_3$) and thermal radiation ($Rd$) on $\theta \left(\eta \right)$ with absolute errors, as shown in Figure 6. Figure 6a depicts the effects of changing the nanoparticles concentration (${\varphi }_1={\varphi }_2={\varphi }_3$) on temperature profiles ($\theta \left(\eta \right)$) with an absolute error ${10}^{-12}\to {10}^{-2}$ as shown in Figure 6b. While Figure 6c shows the impacts of $Rd$ on $\theta \left(\eta \right)$ with absolute error ${10}^{-8}\to {10}^{-4}$ as depicted in Figure 6d. The following impacts on temperature profiles are examined.

• When concentrations of nanoparticles (${\varphi }_1={\varphi }_2={\varphi }_3$) are enhanced then the temperature of fluid raises these phenomena as shown in Figure 6a.
• Thermal radiation has increasing effects on the temperature profile of ternary hybrid nanofluid.

5. Conclusions

The LMS-BPNN approach was used to scrutinize the flow problem of ternary hybrid nanofluid over a porous shrinking sheet due to magnetohydrodynamics (MHD) effects and thermal radiation ($Rd$). By using the appropriate transformation, the dynamical PDEs describing ternary hybrid nanofluid are changed into a system of nonlinear ODEs. The initial data set is prepared by varying Casson fluid parameter ($\beta$), magnetic field parameter ($M$), thermal radiation parameter ($Rd$), and concentrations of nanoparticles with the use of bvp4c (MATLAB built-in function, Lobatto III-A solver). The data sources are used as standard results for 80% of training data, 10% of validation data and 10% of testing data by running the LMS-BPNN solver. The suggested and numerical outcomes validate the technique’s accuracy. The table values and graphical demonstrations of MSE convergence plots, regression analysis, and histogram analysis are shown to show the validity and reliability of the proposed LMS-BPNN. Further, error analysis is conducted to conclude the dependability and accuracy of the data obtained from numerical simulations. The error between numerical values and predicted ones with AI is estimated ${10}^{-6}$.

It is summarized that:

• $f^{\prime }\left(\eta \right)$ decreases with the increment in $\beta$.
• The increment in $f^{\prime }\left(\eta \right)$ is examined when there is an increase in the value of $M$.
• $f^{\prime }\left(\eta \right)$ increases with porosity-increasing effects.
• $\theta \left(\eta \right)$ increases with increasing concentrations of nanoparticles.