Intelligent computing technique based supervised learning for squeezing flow model

In this study, the unsteady squeezing flow between circular parallel plates (USF-CPP) is investigated through the intelligent computing paradigm of Levenberg–Marquard backpropagation neural networks (LMBNN). Similarity transformation introduces the fluidic system of the governing partial differential equations into nonlinear ordinary differential equations. A dataset is generated based on squeezing fluid flow system USF-CPP for the LMBNN through the Runge–Kutta method by the suitable variations of Reynolds number and volume flow rate. To attain approximation solutions for USF-CPP to different scenarios and cases of LMBNN, the operations of training, testing, and validation are prepared and then the outcomes are compared with the reference data set to ensure the suggested model’s accuracy. The output of LMBNN is discussed by the mean square error, dynamics of state transition, analysis of error histograms, and regression illustrations.


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Neural network LMB Levenberg-Marquard backpropagation ρ Fluid density µ Dynamic viscosity w Axial velocities 2ℓ(t) Distance between the plates at any time t Q The volume flow rate p The pressure η Dimensionless variable u Radial velocities ν The velocity of the circular plates Re Reynolds number In fluid dynamics, several areas inspire the researchers to further study and explore applicability and analysis. The flow of squeezing between two parallel circular walls is one of them because of its many valuable and varied applications in our current life reality. The primary vital application is the heart, where it pumps blood to the entire body through pressure. It also has industrial applications and engineering such that injection molding and polymer processing. Stefan 1 publication of a classical study of squeezing flow through the use of lubrication to generate a homogeneous compression provides an aspect to study squeezing flow system. This study is inspired by a series of studies on squeezing flow system investigated by many researchers. Ahmed et al. 2 studied the unsteady squeezing flow considering the viscosity mainly affected by the temperature by applying the killer box method. Çelik et al. 3 investigated the influence of heat transfer and velocity on squeezing flow by the Gegenbauer Wavelet Collocation Method. Sobamowo et al. 4 used both methods of differential transformation and variation of parameters to study the effect of a magnetic field on Casson nanofluid's squeezing flow through a porous medium. Çelik 5 studied the effect of viscosity on squeezing flow in a magnetic field for a specific type of fluid known as Copper-water and Copper-kerosene. Noor et al. 6  www.nature.com/scientificreports/ that helped to analyze the unsteady flow of nanofluid between two disks. Thumma et al. 8 examined the influence of convection on the flow problem of electromagnetohydrodynamic radiative between two circular plates. Some other recent studies that have addressed squeezing flow can be seen in the literature [9][10][11][12][13][14] .
In the previous research, squeezing flow has been studied using different numerical methods, but stochastic numerical computing that is dealing with artificial intelligence is utilized to analyze the fluidic systems recently.
The accurate results provided by stochastic numerical computing have been employed to provide new research in various fields such as fluid mechanics [15][16][17] , biological research 18,19 , business and finance systems 20,21 , models of Panto-graph delay differential systems [22][23][24] , plasma science 25 , thermodynamics 26 , magneto-hydrodynamics 27 , solid conductive materials 28 , atomic physics 29 and other researches of interest.It is worth noting that artificial intelligence is also able to keep pace with modern problems that are emerging in the world in various fields, such as Covid 19 30,31 .
In this study, the system of (USF-CPP) is performed by an intelligent computing paradigm of Levenberg-Marquard backpropagation neural networks (LMBNN). The research proceeds in several steps that can be summarized as follows • Levenberg-Marquard backpropagation neural networks (LMBNN) is developed to discuss the impact of different scenarios connected with the squeezing flow system (USF-CPP). • The governing flow system (USF-CPP) based on partial differential equations (PDEs) is transformed into differential equations (ODEs) for better applicability of networks (LMBNN). • Runge-Kutta method is used to generate a dataset for the USF-CPP problem, which is finally prepared for neural network infrastructure, i.e., LMBNN by variation of Reynolds number and volume flow rate. • LMBNN processes that are testing, training, and validation applied on system presenting the squeezing flow model USF-CPP for various scenarios and cases. • The mean square error discusses the results of LMBNN, dynamics of state transition, analysis of error histograms, and regression illustrations.
The workflow overview of solving USF-CPP with the proposed model LMBNN is presented in (See Fig. 1). The Mathematical formulation of the USF-CPP model exposure in "Solution methodology" section. The present model solution Procedure has been displayed in "Results and discussion" section. The accuracy of the output, the proposed LMBNN, is showing in "Conclusions" section. The conclusion of the research is given in the last section.

Mathematical formulation
The geometry of the squeezing flow of an incompressible two-dimensional viscous fluid between two parallel plates shown in (See Fig. 2). The distance between the two circular plates at any time t is 2ℓ(t) . The speed at which the upper and lower plates move each other is v(t). Select the r-axis as the model's central axis, and the z-axis is normal to it. For axisymmetric flow, assumed that the plates approach symmetrically with respect to r-axis.
The governing system 32 become in form where Subject to the boundary conditions where η = z ℓ(t) , u radial velocity and w axial velocity. To simplify the complex system of differential equations above and make it easier to find and analyze the results, we use similarity transformations and the following equation yields.
where both Re and Q are constant.
The circular plates diverge when Re 0 , while converges towards each other when Re < 0 and the squeezing flow are symmetrical with the velocity profiles, provided ℓ(t) 0 . As well if Q = −Re then Eq.(6) is reduced to

Solution methodology
The Levenberg Marquardt (LM) training technique is an efficient technique in the field of intelligent computing. It is designed to calculate the second-order training fast, and it requires that the output of the neural network operation is a single neuron (See Fig. 3). Implement the Levenberg Marquardt technique in MATLAB based on using the command of the neural network toolbox "nftool" to fit the problem. The total data for LMBNN is 1001 found between 0 and 1 by setting 0.001 as steps, using the Runge-Kutta technique through the "NDSolve" built-in function for numerical solution in Mathematica. The dataset values for f (η) were randomly used for each of the training, validation, and    Fig. 4).

Results and discussion
The numerical application based on LMBNN is presented here for the squeezing flow model obtained in Eqs. (6)(7)(8)(9). The proposed LMBNN is implemented for six scenarios by variation of Re, Q, with three different cases for each scenarios, as shown in Table 1. Notice that the equation associated with value variation is used in each scenario. Figures 5, 6, 7 shows that performance, states, and error histograms for all six scenarios in case 2 for USF-CPP, respectively. Studies of regression are given (See Fig. 8). The fitting of solution respective six scenarios of case 2 is presented (See Fig. 9). Also, LMBNN outcomes are comparing with the standard outcomes (See Figs. 10, 11).

Conclusions
In this paper, the intelligent computing paradigm of Levenberg-Marquard backpropagation neural networks (LMBNN) offered a numerical solution of USF-CPP by simplified the system into an equivalent nonlinear ordinary differential equation with suitable transformation. The Runge-Kutta method is implemented for the USF-CPP dataset by variation of Reynolds number and volume flow rate. The 70% , 15% , and 15% of points are determined for training, testing, and validation for various scenarios of LMBNN. The best agreement of both proposed and reference results along with the level is 10 −06 to 10 −14 . Also, The velocity profile f ′ (η) is directly proportional to the increase of Reynolds number Re and inversely proportional to the volume flow rate Q. Moreover, verifying the scheme accuracy results is achieved through graphs and tables illustrations such as mean square error, state transition dynamics, analysis of error histograms, and regression.
In the future, it will introduce mechanics through new platforms based on artificial intelligence to provide more accurate and efficient results [33][34][35][36] .