EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6795 ISSN 1307-5543 – ejpam.com Published by New York Business Global Prediction of Thermal Behavior in Ferromagnetic Carreau Fluids Using Neural Networks Algorithm Saraj Khan1, Muhammad Imran Asjad1,2,∗, Muhammad Naeem Aslam3, Marei S. Alqarni4, Liliana Guran5 1 Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan 2 Jadara University Research Center, Jadara University, Irbid, Jordan 3 Department of Mathematics, Lahore Garrison University, Lahore, Pakistan 4 Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia 5 Department of Hospitality Services, Faculty of Business, Babes-Bolyai University, Horea Street No. 7, 400174, Cluj-Napoca, Romania Abstract. This study investigates the melting heat transfer characteristics of a ferromagnetic Carreau fluid (FCF) influenced by an external magnetic dipole. The Carreau model captures the non-Newtonian shear-thinning and shear-thickening nature of the fluid, while ferromagnetic ef- fects introduce magnetically induced forces that modify flow and heat transport. The governing nonlinear ordinary differential equations are solved using MATLAB’s bvp4c solver. The resulting reference data are used to train an Artificial Neural Network (ANN) optimized via the Levenberg– Marquardt Technique (LMT). The proposed ANN–LMT framework demonstrates high predictive accuracy with low Mean Squared Error (MSE) values, showing excellent agreement with numerical results. A specific set of physical parameters including the melting parameter (B), ferromagnetic interaction parameter (β), Eckert number (Ec), Prandtl number (Pr), Schmidt number (Sc), dimensionless Curie temperature (ϵ) and Weissenberg number (We) was used to simulate the fer- romagnetic Carreau fluid flow. Physically, increasing We and β reduces the velocity, while higher Pr and B suppress temperature and concentration due to diffusion and melting effects. Con- versely, greater Ec enhances thermal gradients, Sc weakens solute diffusion, and rising ϵ increases concentration. Overall, the ANN–LMT model provides an efficient and accurate computational alternative for analyzing complex magnetothermal flow systems, with potential extensions to un- steady and three-dimensional configurations. 2020 Mathematics Subject Classifications: 76W05, 80A20, 68T07, 76A05 Key Words and Phrases: Levenberg–Marquardt algorithm, ferromagnetic Carreau fluid, ther- mal boundary layer, thermal behavior prediction, magnetohydrodynamics ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6795 Email addresses: saraj.khan.niazi@gmail.com (S. Khan), imran.asjad@umt.edu.pk (M. I. Asjad), muhammadnaeemaslam10@gmail.com (M. N. Aslam), msalqarni@kku.edu.sa (M. S. Alqarni), liliana.guran@ubbcluj.ro (L. Guran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 2 of 33 1. Introduction Ferromagnetic Carreau fluids form a unique class of complex fluids that combine shear dependent viscosity with magnetic responsiveness, enabling control through external magnetic fields [1–3]. This dual behavior, arising from the Carreau model’s non-Newtonian rheology and magnetic susceptibility, has made such fluids valuable in applications like magnetic drug delivery, ferrofluid-based heat exchangers, microfluidics, and cooling sys- tems [4]. The interaction between magnetic forces and nonlinear viscosity introduces ad- ditional complexity in flow modeling and heat transfer analysis. Shah et al. [5] examined MHD convective heat transfer in Carreau fluids and found that temperature-dependent vis- cosity and conductivity significantly affect the thermal boundary layer, while Al-Khafajy et al. [6] analyzed peristaltic transport in elastic channels, revealing key insights into non-Newtonian flow behavior under physiological conditions. Rehman et al. [7] analyzed thermal behavior in converging channels with mass transfer and Cattaneo-Christov heat flux, highlighting the critical role of inertia and heat propa- gation delays in flow modulation. Kutev et al. [8] provided a theoretical investigation into unsteady Carreau fluid flow in pipes, identifying bounds on flow behavior through rheo- logical parameters. Ghosh et al. [9] obtained analytical solutions for ferrofluid convection influenced by a transverse magnetic field, offering a clearer understanding of magnetic damping in boundary layers. Al-Obaidi et al. [10] examined the magnetic stabilization of flow in porous media, indicating that magnetic field strength plays a critical role in flow stability, especially when the Grashof number is significant. The importance of magnetic dipole effects has also been explored in biofluid and non-Newtonian flow models. Dhar- maiah et al. [11] investigated radiative MHD blood flow under the influence of a magnetic dipole, incorporating Brownian motion and thermophoresis. Their results emphasized the significance of magnetic body forces in controlling temperature and velocity profiles in physiological environments. Heat transfer is a critical aspect in ferromagnetic Carreau fluid, as thermal energy gov- erns both phase transitions and the fluid’s magnetic response. Hobiny et al. [12] numer- ically examined viscous dissipation in Carreau fluids, pointing out how shear-dependent viscosity affects thermal energy conversion. In advanced energy systems, Bilal et al. [13] used the parametric continuation method to model energy transfer in Carreau-Yasuda fluids with magnetic dipoles and hybrid nanoparticles. Their study demonstrated that ternary hybrid nanofluids outperform simpler fluids in thermal enhancement. Further, magnetized Carreau fluid is sensitive to mass transfer processes. Diffusion and convection influence the dispersion of magnetic particles, affecting the overall concentration field and functionality of the fluid. Thirupathi et al. [14] modeled stagnation point flow of a Car- reau nanofluid under MHD effects, showing how magnetic and viscous forces reshape flow dynamics. Asha et al. [15] applied the multi-step differential transformation method to examine Joule heating in peristaltic Carreau nanoflows, while Rooman et al. [16] inves- tigated heat transfer and solute transport in renal flow channels using a Carreau model, emphasizing biomedical relevance. The Levenberg–Marquardt technique is a widely recognized optimization method that S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 3 of 33 plays a crucial role in neural network training, particularly for nonlinear regression and curve-fitting tasks. This technique was originally introduced by Kenneth Levenberg in 1944 [17] as a way to solve nonlinear least-squares problems by blending the principles of gradient descent and the Gauss–Newton method. His approach introduced a damping fac- tor to stabilize convergence, which proved particularly effective when the Gauss–Newton method struggled near saddle points or ill-conditioned regions. In 1963, Donald Mar- quardt refined the method [18], improving its stability and performance for estimating parameters in complex nonlinear systems. This combined approach, now referred to as the Levenberg–Marquardt algorithm, provides an effective interpolation between the ro- bustness of gradient descent and the fast convergence of Gauss–Newton. Hagan et al. [19] highlighted the efficiency of the Levenberg–Marquardt technique in training feedforward neural networks, offering fast convergence and high accuracy for medium-sized datasets. Moré [20] noted its robustness in handling complex, ill-conditioned error surfaces, while Huang and Loh [21] demonstrated its strength in nonlinear system modeling. Building on these advantages, recent works have applied LMT-based neural networks to nonlinear fluid dynamics. Albasheir et al. [22] modeled Carreau nanofluid stagnation flow under magnetic and heat generation effects, achieving improved prediction of thermal and solutal gradients. Similar studies on MHD Carreau fluid flows with slip and heat generation confirmed the method’s precision in capturing nonlinear transport behavior [23–26]. Machine learning has become a powerful approach for modeling and optimizing com- plex transport phenomena in non-Newtonian and nanofluid flows. Abbasi et al. [27] used an ML-based framework to predict the 3D behavior of Carreau fluids under Cattaneo– Christov double diffusion theory, while Alhamdi et al. [28] applied supervised learning to enhance heat transfer in bio-convective Carreau blood-based nanofluid flow with nonlin- ear radiation [29, 30]. Priyadharshini et al. [31, 32] optimized ternary hybrid nanofluid and MHD flow using machine learning and gradient descent regression, and Kumar et al. [33, 34] employed deep neural networks and SVMs to predict and classify tri-hybrid nanofluid heat transfer. Baithalu et al. [35] further used a Levenberg–Marquardt neural approach for micropolar nanofluid transport with Cattaneo–Christov flux, emphasizing AI’s growing impact on accurate and efficient thermal-fluid modeling. Ayub et al. [36] de- veloped a neural intelligent model to study heat transfer in tri-hybrid cross bio-nanofluid flow over a wedge, showing high accuracy in predicting nonlinear thermal behavior. Asghar et al. [37] applied an ANN approach to solve a nonlinear Cassava mosaic disease model, demonstrating the robustness of ANN techniques in capturing complex biological dynamics. Hassan et al. [38] investigated heat transfer and entropy generation in ferrofluids under a low oscillating magnetic field and reported significant effects of magnetic oscillations on thermal irreversibility. Rizwan et al. [39] developed a rheological model for metallic oxide nanoparticles dispersed in non-Newtonian nanofluids and explored their heat and mass transfer characteristics, revealing enhanced thermal performance due to nanoparticle interactions and flow behavior. Arif et al. [40] investigated cross-diffusion in MHD Williamson nanofluid flow over a nonlinear stretching surface using Morlet wavelet neural networks, demonstrating the method’s accuracy in capturing nonlinear thermal and S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 4 of 33 concentration effects. Alraddadi et al. [41] analyzed a ternary hybrid cross bio-nanofluid in an expanding/contracting cylinder under an inclined magnetic field, showing improved heat transfer. Shah et al. [42] proposed an efficient numerical scheme for melting energy transport in time-dependent Carreau nanofluids. Shah et al. [43] further examined inclined MHD tri-hybrid bio-nanofluid flow using an ANN approach. Despite extensive research on non-Newtonian fluids and magnetohydrodynamic (MHD) effects, limited attention has been paid to the integration of ferromagnetic behavior with phase change phenomena in Carreau-type fluids. Additionally, few studies have addressed the computational challenges of solving such highly nonlinear boundary layer problems using intelligent data-driven strategies. Motivated by this gap, the present work explores the melting heat transfer characteristics of ferromagnetic Carreau fluid (FCF) under the influence of a magnetic dipole field. A novel hybrid methodology is proposed that combines the traditional bvp4c numerical scheme with the Levenberg–Marquardt backpropagation technique (LMT), enabling accurate approximation of velocity, temperature, and concen- tration profiles. The application of LMT to this class of FCF problems introduces a robust predictive framework, offering enhanced accuracy, faster convergence, and deeper insight into the coupled thermal–solutal behavior of magnetized non-Newtonian systems. The primary objective of this study is to develop an efficient and accurate computa- tional framework for analyzing the melting heat transfer and flow characteristics of a FCF under the influence of an external magnetic dipole. To achieve this, the nonlinear boundary value problem governing the FCF is formulated and solved using MATLAB’s bvp4c solver to obtain reliable reference data for velocity, temperature, and concentration profiles. An artificial neural network trained with the LMT is then employed to approximate these reference solutions and evaluate the model’s predictive accuracy. The ANN–LMT pre- dictions are compared with the numerical results to validate the robustness, convergence, and generalization capability of the proposed hybrid modeling approach. This method- ology effectively bridges numerical precision and data-driven learning, offering a reliable framework for simulating nonlinear magnetothermal systems influenced by melting and diffusion effects. The present study is guided by several research questions that define its scope and objectives. It investigates how effectively a supervised neural network optimized with the Levenberg–Marquardt algorithm can predict the nonlinear thermal and flow behavior of ferromagnetic Carreau fluids. It also explores the combined effects of magnetic interaction, Weissenberg number, and Prandtl number on the velocity and temperature profiles of such fluids. Finally, it examines how accurately the proposed ANN–LMT framework can reproduce the results obtained from conventional numerical solvers such as MATLAB’s bvp4c. The structure of the paper is outlined as follows: Section 1 presents a detailed review of the relevant literature along with the physical context underlying the study. Section 2 formulates the mathematical model by introducing the governing equations and corre- sponding boundary conditions. In Section 3, the implementation of an ANN-LMT for modeling the ferromagnetic Carreau fluid is discussed. Section 4 provides graphical re- sults and a comprehensive discussion of the observed physical behavior. Finally, Section 5 S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 5 of 33 summarizes the main conclusions of the study. 2. Problem Modeling Figure 1 shows the two-dimensional flow of a ferromagnetic nanofluid over an imperme- able sheet of a stretched sheet horizontally. A magnetic field is applied on the outside to affect the flow. The x-axis has two equal and opposite forces, and the y-axis is perpendic- ular to the surface of a sheet. The sheet is above a fixed distance of a y-axis by a magnetic dipole of known strength with a fixed field in the positive x-direction. This generates a magnetic field owing to the dipole; this boosts the magnetic field concentration locally, and ultimately the magnetic field effects lead to magnetic saturation in the ferrofluid. Figure 1: Flow geometry The temperature of the sheet, which we indicate by Tw, is kept below the Curie temper- ature Tc. At temperatures beyond this magnetization limit, the ferromagnetic nanoparti- cles lose their magnetism and end up becoming paramagnetic particles. It is assumed that the fluid far from the sheet has a temperature T∞ = Tc, meaning the nanoparticles remain non-magnetic until they cool upon entering the thermal boundary layer. A mathematical model is formulated to examine the influence of magnetic dipoles on a generalized New- tonian (Carreau) nanofluid, incorporating the effects of viscous dissipation and chemical reactions. The governing equations describing the conservation of mass, momentum, energy, and concentration for a two-dimensional, incompressible, steady flow of a ferromagnetic Car- reau fluid are first expressed in their general vector form as follows [44]: ∇ ·V = 0, ρ(V · ∇)V = −∇p+∇ · τ + Fm, ρcp(V · ∇T ) = k∇2T +QT , S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 6 of 33 (V · ∇C) = DB∇2C + DT Tc ∇2T. Here, V = (ū, v̄) denotes the velocity vector, ρ is the fluid density, p is the pressure, T represents the temperature, and C denotes the nanoparticle concentration. The extra stress tensor τ for the Carreau fluid model is defined as τ = 2µeff(γ̇)D, D = 1 2 ( ∇V + (∇V)T ) , where D is the rate of deformation tensor and µeff(γ̇) is the effective viscosity given by the Carreau relation, µeff(γ̇) = µ [ 1 + (Γγ̇)2 ]n−1 2 , with γ̇ = √ 2D : D being the shear rate. The magnetic body force Fm arises due to the spatial variation of magnetization and magnetic field intensity, and for a temperature-dependent magnetization M(T ), it is ex- pressed as Fm = µ0M(T )∇H, where µ0 denotes the magnetic permeability, M(T ) is the magnetization, and H represents the magnetic field strength. The term QT in the energy equation accounts for the thermal effects induced by mag- netization and can be written as QT = µ0 T dM dT (V · ∇H), which incorporates the coupling between the temperature gradient, magnetization, and the applied magnetic field. The last equation represents the nanoparticle concentration field, which includes both Brownian motion and thermophoretic diffusion effects through the coefficients DB and DT , respectively. The above vector equations can be expressed explicitly in their two-dimensional com- ponent form as: ∂v̄ ∂y + ∂ū ∂x = 0, (1) ∂ū ∂y v̄ + ∂ū ∂x ū = −1 ρ ∂p ∂x + 3ν(n− 1) 2 Γ ( ∂ū ∂y )2 ∂2ū ∂y2 + µ0M ρ ∂H ∂x , (2) ∂T ∂y v̄ + ∂T ∂x ū+ µ0 ρcp T ∂M ∂T ( ū ∂H ∂x + v̄ ∂H ∂y ) = k ρcp ∂2T ∂y2 , (3) S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 7 of 33 ∂C ∂x ū+ ∂C ∂y v̄ = DT Tc ∂2T ∂y2 +DB ∂2C ∂y2 . (4) The boundary conditions suitable for solving the governing equations are expressed as follows:  y = 0 : v̄ = 0, ū = uw = ax, k ( ∂T ∂y ) = ρv̄(0) [Cs(Tw − T0) + λ1] , C = Cw = Cc − B1x L , T = Tw, y → ∞ : v̄ = 0, ū→ 0, T → Tc, C → C∞ = Cc − D1x L . (5) It is worth noting that the melting boundary condition in Eq. 5 unifies the effects of heat flux and melting within a single formulation. This treatment is based on the interfacial energy balance principle, wherein the latent heat absorbed at the solid–liquid interface is directly related to the temperature difference across the boundary. The present boundary expression is consistent with the formulation described by Endalew and Sarkar [44], ensur- ing that the thermal energy absorbed due to melting is properly coupled with conductive heat transfer at the surface. The parameters that are used in the governing equations are λ1 as the latent heat of the nanofluid, T0 as the temperature of the solid surface, and Cs as the heat capacity of the solid surface. The model parameters, B1 and D1, are both positive constants, whereas the characteristic length scale is L = √ ν/a, where a is a stretching rate constant and ν is the kinematic viscosity. The effect of the magnetic field on the FCF flow is presented with the help of the magnetic dipole. The magnetic scalar potential that corresponds to this dipole can be written as [45]: Φ = α1 2π ( x x2 + (y + c̄1)2 ) . (6) The parameter α1 is magnetic field strength at the source, and components of a mag- netic field vector H are denoted as follows: Hx = ∂Φ ∂x = α1 2π [ x2 − (y + c̄1) 2 (x2 + (y + c̄1)2) 2 ] (7) Hy = ∂Φ ∂y = α1 2π [ 2x(y + c̄1) (x2 + (y + c̄1)2) 2 ] (8) S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 8 of 33 The magnetic body force being proportional to the gradient of the magnetic field magni- tude H, it is necessary that: H = [( ∂Φ ∂x )2 + ( ∂Φ ∂y )2 ]1/2 , (9) ∂H ∂x = α1 2π ( 2x (y + c̄1)4 ) , ∂H ∂y = α1 2π ( − 2 (y + c̄1)3 + 4x2 (y + c̄1)5 ) . (10) An approximate linear relation is employed to express the dependence of magnetization M on temperature T as: M = −K(T − Tc) (11) A parameter known as the pyromagnetic coefficient is indicated by the symbol K. For ferrohydrodynamic interaction to occur, two essential conditions must be satisfied: (i) the fluid temperature T must remain below the Curie temperature Tc, that is, Tc > T ; and (ii) a nonhomogeneous magnetic field must be applied. Ferrofluid must be heated up to its Curie temperature Tc to guarantee that no more magnetization will occur. At a temperature of this, the ferrofluid will stay closer to the surface, as seen through equation (11). The equations that govern the FCF flow model are also non-dimensionalized by intro- ducing the following dimensionless quantities:  ψ(η) = x √ νaf, η = √ a ν y, u = ∂ψ ∂y , v = −∂ψ ∂x , θ(η) = Tc − T Tc − Tw , ϕ(η) = Cc − C Cc − Cw (12) The use of the previously defined stream function automatically satisfies the continuity equation (1). By employing an appropriate similarity transformation, the partial differ- ential equations (2)–(4) are reduced to a system of highly nonlinear and coupled ODEs. ( 1 + 3(n− 1) 2 We2f ′′2 ) f ′′′ − f ′2 + ff ′′ − 2βθ (η + α)4 , (13) θ′′ + Pr(fθ′ − f ′θ)− Pr Ecβ(θ − ϵ) ( 2f ′ (η + α)4 + 4f (η + α)5 ) + λβ(θ − ϵ) 2f (η + α)3 = 0 (14) S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 9 of 33 ϕ′′ + Sc(fϕ′ − f ′ϕ) = 0 (15) The resulting boundary conditions, which were obtained using the similarity transfor- mation, are as given below: { f ′(0) = 1, θ(0) = 1, Bθ′(0) + Prf(0) = 0, ϕ(0) = 1, f ′(∞) = 0, θ(∞) = 0, ϕ(∞) = 0. (16) The physical parameters describing the flow are the following:  Pr = ρcp νk , β = µ0Kρα1(Tc − Tw) 2πµ2 , λ = µ2a akp , α = √ c̄12ρa µ , B = Cp(Tc − Tw) Cs(T0 − Tw) + λ1 , Ec = u2w cp(Tc − Tw) , Sc = ν D , We = Γ a, ϵ = Tc Tc − Tw (17) The dimensionless parameters used are as follows: Pr the Prandtl number, the ferro- hydrodynamic interaction parameter is denoted by β, the viscous dissipation parameter is defined as λ, and the non-dimensional distance between the origin and the center of the magnetic pole is set as α. As well, B is the melting parameter, Ec is the Eckert number, Sc is the Schmidt number, We is the Weissenberg number, and the dimensionless Curie temperature is ϵ. 3. Artificial Neural Network Modeling of FCF using LMT A feedforward artificial neural network (ANN) is employed to approximate the nonlin- ear behavior of ferromagnetic Carreau fluid (FCF), trained using the Levenberg–Marquardt technique (LMT). This hybrid approach combines the rapid convergence of the Gauss– Newton method with the robustness of gradient descent, offering an efficient framework for solving the governing differential equations. The LMT algorithm utilizes a damped least-squares optimization scheme. The weights w are updated iteratively as: wk+1 = wk − ( J⊤J+ µI )−1 J⊤e, where: • The Jacobian matrix of partial derivatives of network error with respect to weights is denoted as J, S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 10 of 33 • e is the error vector between predicted and target outputs, • µ is the damping factor, • I is the identity matrix. The term J⊤J approximates the Hessian matrix, while J⊤e represents the gradient. This formulation ensures stable and accurate training of the ANN for capturing the flow behavior under complex physical conditions. In this study, the input features include spatial variables and physical parameters such as B, β, Ec, Pr, Sc, ϵ, and We, while the target outputs are velocity, temperature, and concentration values obtained using MATLAB’s bvp4c solver. These datasets are then fed into a supervised neural network with the following configuration: • Architecture: One hidden layer with 10 neurons. • Activation: The hidden layer uses the tanh (hyperbolic tangent sigmoid) activation function, suitable for smooth approximation of nonlinear mappings. • Training Algorithm: trainlm, MATLAB’s implementation of LMT. • Data Division: 70% training, 15% testing and 15% validation. • Performance Measure: Mean squared error (MSE). The training process is executed using either MATLAB’s GUI-based nftool Graphical User Interface. This hybrid LMT-ANN approach demonstrates fast convergence and high precision in capturing the complex dynamics of magnetized non-Newtonian flow, as re- flected in the convergence metrics and predictive accuracy summarized in Table 1. 4. Graphical Results and Discussion Table 1: LMT performance metrics for different physical parameters in FCF flow. Parameter Label MSE Epoch Elapsed Time PerformanceGradient Mu Training ValidationTesting B a 1.44×10−8 1.46×10−8 2.03×10−8 711 03 sec 1.23×10−8 1.00×10−7 1.00×10−9 β b 2.07×10−9 1.97×10−9 2.57×10−9 763 01 sec 3.11×10−9 9.43×10−8 1.00×10−8 Ec c 2.17×10−8 3.34×10−8 3.05×10−8 352 01 sec 2.10×10−8 9.98×10−8 1.00×10−8 Pr d 1.85×10−9 2.10×10−9 3.10×10−9 507 02 sec 2.33×10−9 9.94×10−8 1.00×10−7 We e 7.21×10−9 9.15×10−9 8.72×10−9 500 02 sec 9.91×10−9 9.96×10−8 1.00×10−7 ϵ f 1.34×10−9 1.79×10−9 2.16×10−9 585 02 sec 2.00×10−9 9.95×10−8 1.00×10−8 Sc g 1.52×10−9 1.68×10−9 1.88×10−9 84 01 sec 1.76×10−9 5.97×10−7 1.00×10−9 Table 1 compiles the convergence outcomes of the Levenberg-Marquardt technique, including mean squared error (MSE) for training, validation, and testing phases, number of S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 11 of 33 Figure 2: Architecture of the neural network model trained using the LMT for FCF flow. epochs, elapsed training time, gradient values, and the backpropagation control parameter Mu. These results confirm the accuracy and reliability of the LMT-based neural network model across various physical parameters. The LMT is employed to iteratively adjust the network weights, effectively minimizing prediction errors and ensuring high accuracy. Figure 2 presents the neural network architecture and outlines the core mechanism of the Levenberg–Marquardt training process. The architecture of a feedforward neural network trained using the Levenberg-Marquardt technique is illustrated in Figure 3. The input layer consists of three neurons, denoted as x1, x2, and x3, which represent the input features of the system. These input neurons are fully connected to a single hidden layer comprising ten neurons, labeled as h1, h2, . . . , h10. Each hidden neuron employs the tanh activation function, enabling nonlinear transforma- tions of the inputs to enhance the network’s learning capability. The output layer contains a single neuron, labeled y, which produces the final output of the network. All inter-layer connections are fully weighted and adjusted during training using the LMT, a widely used optimization method for training small- to medium-sized neural networks. Figure 4 presents the flow diagram of the ferromagnetic Carreau fluid (FCF) model along with the integrated Levenberg-Marquardt training procedure. The diagram outlines the interaction between flow, thermal, and concentration fields under the influence of magnetic forces and melting effects. It also demonstrates the use of numerical data from the bvp4c solver to train the neural network, where the LMT algorithm iteratively updates weights to minimize prediction errors and enhance learning efficiency. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 12 of 33 x1 x2 x3 h1 tanh h2 tanh h3 tanh h4 tanh h5 tanh h6 tanh h7 tanh h8 tanh h9 tanh h10 tanh yInput Layer Hidden Layer (tanh) Output Layer Figure 3: Feedforward neural network structure trained using the LMT. This study employs the Levenberg-Marquardt technique (LMT) to model the melting heat transfer and flow behavior of a ferromagnetic Carreau fluid subjected to an external magnetic dipole. The governing boundary layer equations, described in Eqs. 13–16, ac- count for the shear-dependent viscosity of the Carreau model and the magnetic body forces due to ferromagnetic properties. MATLAB’s bvp4c numerical solver provides benchmark solutions against which the neural network results are compared. The input domain is defined as η ∈ [0, 4], discretized into 801 points using a step size of 0.005. For each dis- cretized point, the dependent variables include the velocity f(η), temperature θ(η), and concentration ϕ(η), obtained through the numerical solutions of the governing equations. The reference dataset generated using MATLAB’s bvp4c solver therefore comprised 801 data samples for each of the seven governing parameters (B, β,Ec,Pr, Sc, ε,We), resulting in a total of (801 × 7) = 5607 input–output pairs. Each data instance contained seven input features and three corresponding target variables representing the velocity, temper- ature, and concentration profiles. The dataset was divided into 70% for training, 15% for validation, and 15% for testing to ensure robust generalization. The reported Mean Squared Error (MSE) values in Table 1 correspond to the testing phase, reflecting the unbiased predictive accuracy of the Levenberg–Marquardt neural network model. The tuples of the artificial neural network are fed through three layers, where the single hidden layer employs the hyperbolic tangent activation function, commonly referred to as tanh. The activation function is defined as tanh(x) = ex − e−x ex + e−x = 2 1 + e−2x − 1, (18) which is a smooth and differentiable function that maps real-valued inputs to the range (−1, 1). The tanh function is particularly suitable for nonlinear regression and function approximation problems because of its symmetry around zero and its ability to capture smooth nonlinear transitions. This symmetric range keeps neuron activations centered near zero, reducing bias shifts and improving gradient flow during backpropagation. As a result, the training process achieves faster convergence and avoids gradient saturation S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 13 of 33 issues that often occur with other sigmoidal functions. The smooth differentiability of the tanh function also enhances the network’s capacity to learn complex nonlinear dependen- cies between the physical parameters and the corresponding velocity, temperature, and concentration profiles, thereby improving the predictive accuracy of the model. Figures 5, 6, and 7 indicate the LMT-based data analysis for the melting parameter (B), the ferromagnetic interaction parameter (β), and the Eckert number (Ec) respectively. Figures 5(a)–11(a) display the LMT validation performance outcomes for the complete dataset under different parametric settings. Figures 5(b)–11(b) illustrate the evolution of the gradient, Mu parameter, and validation checks throughout training. Figures 8, 9, 10, and 11 demonstrate the LMT response for the Prandtl number (Pr), the Weissenberg number (We), the Curie temperature (ϵ), and the Schmidt number (Sc) respectively. The corresponding error histograms that reflect the network’s prediction accuracy are shown in Figures 5(c)–11(c), revealing that the prediction errors remain closely centered around zero. Regression analysis findings are displayed in Figures 5(d)–11(d), confirming strong correlation between the predicted and target outputs across all settings. Figures 5(a)–11(a) show the convergence plots of the MSE for the test, training, and validation curves across all parametric settings, indicating that the best performance is achieved at [711, 763, 352, 507, 500, 585, 84] epochs, with MSE values around [1.2347 × 10−8, 3.1095×10−9, 2.1024×10−8, 2.3325×10−9, 9.9117×10−10, 1.9956×10−10, 1.7624× 10−9] respectively. The gradient and Mu parameter values for the Levenberg-Marquardt technique are [9.9983 × 10−8, 9.4343 × 10−8, 9.9755 × 10−8, 9.9434 × 10−8, 9.9612 × 10−8, 9.947×10−8, 5.9727×10−7] and [1×10−9, 1×10−8, 1×10−8, 1×10−7, 1×10−7, 1× 10−8, 1 × 10−9] respectively, as illustrated in Figures 5(b)–11(b). These results confirm the convergence efficiency and accuracy of the LMT-based neural network approach. The error histograms in Figures 5(c)–11(c) further corroborate these findings, showing that the difference between network output and reference data remains close to zero. When compared to the reference zero-line error bin, the histograms reveal errors of approximately [−2.1×10−5, 3.54×10−6, 3.91×10−5, 7.53×10−6, 4.11×10−6, −6.1×10−7, −9×10−7] respectively. Figures 5(d)–11(d) illustrate the regression analysis for the ferromagnetic Carreau fluid flow model with melting effects under various parametric configurations. The regression plots reveal strong correlations (R values close to 1), confirming the predictive reliability and accuracy of the neural network trained using the Levenberg-Marquardt technique. Additionally, the limited amount of scatter and the nearly perfect alignment of predicted versus target data suggest the absence of significant errors or missing values in the modeled system. As illustrated in Figures 5(e)–11(e), the fitness graphs throughout the training, validation, and testing phases demonstrate prediction errors in the range of 10−5 per unit time. Targets are denoted by dots (·), whereas predicted outcomes for training, validation, and testing are shown by plus signs (+). Figure 12 illustrates the effect of the Weissenberg number (We) on the velocity profile for shear-thinning fluids, characterized by a power-law index n < 1. It is evident that the velocity decreases as We increases. The Weissenberg number quantifies the ratio of elastic to viscous effects in a non-Newtonian fluid. An increase in We implies a longer relaxation time, which enhances the elastic response of the fluid. This elevated elasticity S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 14 of 33 increases the internal resistance to deformation, thereby suppressing the flow and reducing the fluid velocity. The effect is more pronounced in shear-thinning fluids, where viscosity decreases with shear rate, yet elastic stresses continue to resist motion. Figure 13 shows the effect of the ferromagnetic interaction parameter β on the velocity profile. As β increases, the velocity decreases due to the enhanced magnetic field generated by the dipole. This magnetic field introduces a resistive (drag) force that opposes fluid motion, thereby slowing down the flow. The greater the magnetic influence, the stronger the suppression of velocity through magnetically induced resistance. Figure 14 shows that an increase in the Prandtl number (Pr) reduces the thermal boundary layer thickness for both fluids. Physically, higher Pr indicates lower thermal diffusivity, limiting heat transfer. As a result, the temperature gradient near the surface increases, and the thermal boundary layer becomes thinner than the momentum boundary layer. Figure 15 illustrates the effect of the Weissenberg number (We) on the thermal distribution for both fluid types. For shear-thinning fluids, increasing We enhances the thermal boundary layer thickness due to stronger elastic effects, which promote energy retention. In contrast, shear-thickening fluids show a reduction in thermal boundary layer thickness with higher We, as increased viscosity dominates and restricts thermal diffusion. Figure 16 illustrates the effect of the ferromagnetic interaction parameter β on the temperature profile. This parameter represents the strength of the magnetic field generated by an external dipole. As β increases, the intensified magnetic field imposes a stronger restrictive force on fluid motion, reducing convective mixing and suppressing the fluid’s ability to transport thermal energy. As a result, thermal diffusion weakens and the temperature profile decreases for both shear-thinning and shear-thickening fluids. This demonstrates how increased magnetic interaction can inhibit heat transfer by limiting fluid deformation and internal energy distribution. Melting refers to the phase transition from solid to liquid due to heat absorption. During this process, energy is drawn from the fluid layers adjacent to the surface within the thermal boundary layer. As shown in Figure 17, the inclusion of the melting parameter B reduces the temperature gradient near the wall. This occurs because the heat required for melting lowers the available thermal energy for conduction into the fluid, thereby weakening the thermal boundary layer. Consequently, the surface temperature rises more slowly, and the overall thermal transport is moderated. Figure 18 illustrates the influence of viscous dissipation, represented by the Eckert number (Ec), on the temperature profile. As Ec increases, a noticeable reduction in the thermal boundary layer thickness is observed. Physically, a higher Ec implies more conversion of kinetic energy into internal energy due to viscous effects. However, in this case, the generated heat is localized and does not significantly spread through conduction, resulting in a sharper thermal gradient near the surface and a thinner thermal boundary layer. Figure 19 presents the effect of the Schmidt number (Sc) on the concentration pro- file. An increase in Sc indicates a lower mass diffusivity relative to momentum diffusivity, meaning solute particles diffuse more slowly through the fluid. As a result, the concen- tration boundary layer becomes thinner, and the overall concentration within the fluid decreases. This behavior reflects reduced mass transport due to limited molecular diffu- S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 15 of 33 sion, especially in fluids with higher viscosity or lower solute mobility. Figure 20 illustrates the effect of the Weissenberg number (We) on the concentration profile. As We increases, the concentration gradient decreases, leading to a lower solute concentration throughout the domain. This behavior is attributed to the increased elastic effects associated with higher We, which enhance fluid resistance and suppress convective transport. Conse- quently, the reduced mobility of solute particles weakens mass diffusion, resulting in a thinner concentration boundary layer. The effect of the dimensionless Curie temperature (ϵ) on the concentration profile is presented in Figure 21. The results indicate that the concentration increases with rising ϵ. Physically, an increase in ϵ implies that the system temperature is approaching the Curie temperature, beyond which the ferromagnetic material gradually loses its magnetic properties. As the magnetic field weakens, the associated magnetic body forces that typically aid in the outward diffusion of solute particles diminish. Consequently, the suppression of magnetic convection leads to reduced solute dispersion away from the wall, allowing more solute particles to accumulate in the near-wall region. This results in an enhanced concentration profile within the boundary layer. The effect of the melting parameter B on the concentration profile is illustrated in Figure 22. As B increases, it enhances the heat transfer from the solid surface into the fluid, promoting localized melting and increased thermal energy near the boundary. This elevated thermal activity leads to enhanced mixing and reduces the accumulation of solute near the surface. Consequently, the concentration profile decreases due to the dilution of solute particles and the disruption of the concentration boundary layer. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 16 of 33 The system of nonlinear PDEs governing the ferromagnetic Carreau fluid flow was transformed into a coupled nonlinear system of ordinary differential equations using similarity transformations. Modeling of ferromagnetic Carreau fluid flow LMT-based neural networks are applied to approximate velocity, temperature, and concentration profiles in ferromagnetic Carreau fluid. Designed LMT for ferromagnetic Carreau fluid flow Model LMT-based outcomes utilizing reference data for the ferromagnetic Carreau fluid The bvp4c solver produced reference datasets by varying key parameters of the ferromagnetic Carreau fluid, supporting LMT training and validation. Normalize and organize numerical data for supervised neural network input. Train ANN using Levenberg–Marquardt algorithm to minimize prediction error. Figure 4: Flow diagram representing the ferromagnetic Carreau fluid model. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 17 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 5: Effect of the melting parameter (B) on LMT-based results S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 18 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 6: Effect of the ferromagnetic interaction parameter (β) on LMT-based results S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 19 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 7: Effect of the Eckert number (Ec) on LMT-based results S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 20 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 8: Effect of the Prandtl number (Pr) on LMT-based results S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 21 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 9: Effect of the Weissenberg number We on LMT-based results. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 22 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 10: Effect of the Curie temperature ϵ on LMT-based results. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 23 of 33 (a) Performance Plot (b) Training State Plot (c) Error Histogram (d) Regression Plot (e) Function fitting graph Figure 11: Effect of the Schmidt number (Sc) on LMT-based results. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 24 of 33 Figure 12: The velocity profile affected by the Weissenberg number (We). Figure 13: The velocity profile affected by the fer- rohydrodynamic interaction parameter (β). Figure 14: Effect of Weissenberg number (We) on temperature profile. Figure 15: Effect of Prandtl number (Pr) on temper- ature profile. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 25 of 33 Figure 16: Temperature affected by ferrohydrody- namic interaction parameter (β). Figure 17: Influence of the melting parameter (B) on temperature profile. Figure 18: Effect of Eckert number (Ec) on temperature profile. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 26 of 33 Figure 19: Effect of Eckert number (Sc) on concen- tration profile. Figure 20: Effect of Weissenberg number (We) on concentration profile. Figure 21: Influence of Curie temperature (ϵ) on con- centration profile. Figure 22: Influence of melting parameter (B) on con- centration profile. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 27 of 33 5. Conclusion This study presented a hybrid computational framework combining numerical simu- lation using MATLAB’s bvp4c solver and the Levenberg–Marquardt Technique (LMT) to investigate the melting heat transfer and flow behavior of a ferromagnetic Carreau fluid (FCF) influenced by an external magnetic dipole. The effects of key dimensionless parameters, including the melting parameter (B), ferromagnetic interaction parameter (β), Eckert number (Ec), Prandtl number (Pr), Schmidt number (Sc), and dimensionless Curie temperature (ϵ), were comprehensively examined to understand their impact on the flow and thermal characteristics of the system. • The dataset was divided into 70% for training, 15% for validation, and 15% for testing to ensure robust generalization. • The LMT-based Artificial Neural Network (ANN) exhibited strong predictive per- formance, achieving extremely low Mean Squared Error (MSE) values on the order of 10−8 to 10−10 and high correlation coefficients (R ≈ 1) across all parameter con- figurations. • Optimal convergence occurred at epochs [711, 763, 352, 507, 500, 585, 84], corre- sponding to MSE values of approximately [1.2347 × 10−8, 3.1095 × 10−9, 2.1024 × 10−8, 2.3325× 10−9, 9.9117× 10−10, 1.9956× 10−10, 1.7624× 10−9]. • Error histograms indicated a narrow deviation range [−2.1×10−5, 3.54×10−6, 3.91× 10−5, 7.53× 10−6, 4.11× 10−6, −6.1× 10−7, −9× 10−7], confirming the network’s precision and stability. • Regression plots and error distributions validated the ANN’s reliability, showing that prediction errors were tightly clustered around zero and closely aligned with the reference data. • Increasing We reduced velocity and concentration due to elastic effects, while tem- perature responded oppositely for shear-thinning and shear-thickening fluids—indicating energy retention in shear-thinning and suppression in shear-thickening regimes. • A higher ferromagnetic parameter (β) decreased both velocity and temperature, demonstrating that strong magnetic interactions hinder convective motion and mo- mentum transfer. • The melting parameter (B) lowered temperature and concentration due to latent heat absorption, weakening energy diffusion and reducing near-wall gradients. • Larger Pr values thinned the thermal boundary layer as a result of reduced thermal diffusivity, whereas higher Ec values intensified thermal gradients near the wall. • An increase in Sc weakened solute diffusion, and higher ϵ enhanced near-wall con- centration due to diminished magnetic forces near the Curie temperature, reducing solute dispersion. S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 28 of 33 Overall, the findings confirm the robustness and predictive capability of the ANN– LMT hybrid framework in modeling complex nonlinear ferrohydrodynamic systems in- volving melting and mass diffusion. The proposed approach offers a computationally efficient and accurate alternative to purely numerical solvers, with significant potential for practical applications in magnetic cooling, materials processing, polymer extrusion, and bioengineering heat transfer systems. Future studies may focus on unsteady-state and three-dimensional geometries, exper- imental validation, and the integration of advanced neural architectures such as Physics- Informed Neural Networks (PINNs) to further enhance modeling accuracy, adaptability, and physical consistency. Table 2: Nomenclature defining symbols, dimensionless parameters, and abbreviations used in the study. Symbol Description Symbol Description u, v Velocity components along x, y ū, v̄ Dimensional velocity components T Fluid temperature (K) Tw, Tc, T∞ Wall, Curie, and ambient temperatures (K) C Solute concentration (mol/m3) Cw, C∞ Wall and ambient concentrations (mol/m3) ρ Fluid density (kg/m3) µ, ν Dynamic and kinematic viscosity k Thermal conductivity (W/m·K) cp Specific heat at constant pressure (J/kg·K) µ0 Magnetic permeability (H/m) M,H Magnetization and magnetic field strength DB , DT Brownian and thermophoretic diffusion coeff. K Pyromagnetic coefficient α1 Magnetic field constant at source c̄1 Distance between dipole and surface L Characteristic length (m) a Stretching rate constant D Rate of deformation tensor B1, D1 Constants Cs Heat capacity of solid surface λ1 Latent heat of nanofluid B Melting parameter β Ferrohydrodynamic interaction parameter Ec Eckert number Pr Prandtl number Sc Schmidt number We Weissenberg number ε Dimensionless Curie temperature λ Viscous dissipation parameter α Non-dimensional magnetic dipole distance n Power-law index f(η) Dimensionless velocity profile θ(η) Dimensionless temperature profile ϕ(η) Dimensionless concentration profile η Similarity variable θ(η) Dimensionless temperature profile MSE Mean Squared Error ANN Artificial Neural Network LMT Levenberg–Marquardt technique Table 2 presents a comprehensive overview of all symbols, parameters, and abbrevia- tions used in the analysis, ensuring clarity and consistency throughout the manuscript. Funding Statement This study received no particular support from public, corporate, or nonprofit entities. Declaration of Competing Interest The authors declare that none of their personal ties or known conflicting financial interests might have seemed to have influenced the work described in this study. Acknowledgement The authors are thankful to the University of Management and Technology Lahore Pak- istan for facilitating to support this research. Data Availability S. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6795 29 of 33 The corresponding author can provide the datasets that were utilized in this work upon reasonable request. References [1] Muhammad Tabrez, Waqar Azeem Khan, Taseer Muhammad, Iftikhar Hussain, and Muhammad Waqas. Significance of thermo-dynamical moment of ferromagnetic nanoparticles and bioconvection analysis for magnetized carreau fluid under the in- fluence of gyrotactic moment of microorganisms. Tribology International, 186:108633, 2023. 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