Novel Adaptive Bayesian Regularization Networks for Peristaltic Motion of a Third-Grade Fluid in a Planar Channel

Abstract

In this presented communication, a novel design of intelligent Bayesian regularization backpropagation networks (IBRBNs) based on stochastic numerical computing is presented. The dynamics of peristaltic motion of a third-grade fluid in a planar channel is examined by IBRBNs using multilayer structure modeling competency of neural networks trained with efficient optimization ability of Bayesian regularization method. The reference dataset used as inputs and targets parameters of IBRBN has been obtained via the state-of-the-art Adams numerical method. The data of solution dynamics is created for multiple scenarios of the peristaltic transport model by varying the volume flow rate, material parametric of a third-grade fluid model, wave amplitude, and inclination angles. The designed integrated IBRBNs are constructed by exploiting training, testing, and validation operations at each epoch via optimization of a figure of merit on mean square error sense. Exhaustive simulation of IBRBNs with comparison on mean square error, histograms, and regression index substantiated the precision, stability, and reliability to solve the peristaltic transport model.

1. Introduction

Peristaltic transport systems of fluid dynamics studies have utmost interest in the research community due to occurrence in the esophagus, the ureter, and the lower intestine models. Peristaltic transport models of mechanical pumps are also normally used for the prevention of fluid contamination or corrosion in the moving parts of the pumps’. A lot of analytical and numerical studies are conducted for peristaltic flows of Newtonian and non-Newtonain fluids such as peristaltic pumping with Newtonian fluids is reported in [1], peristaltic pumping model for the circular cylindrical tubes [2], temperature effects for the electroosmosis modulated peristaltic transport model [3], peristaltic motion dynamics of the suspension in the small intestine model [4], peristaltic transport model with multilayered power-law fluidic system [5], peristaltic motion of Rabinowitsch liquid [6], peristaltic transport of two-layered blood model [7], numerical study of peristaltic transport of the thixotropic fluids [8], peristaltic transport of a Rabinowitsch fluid [9], peristaltic transport of Casson fluid [10], peristaltic motion of the Eyring-Powell fluid [11], peristaltic transport of a micropolar nanofluidic system [12], radiative peristaltic motion of Eyring-Powell fluid mixed with nanoparticles [13], asymmetric peristaltic propulsion model [14], peristaltic motion of the viscoelastic physiological fluids [15], peristaltic transport of Jeffrey fluid in the presence of high magnetic field [16], peristaltic transport of Bingham nanofluid [17], peristaltic transport of hybrid nanomaterial based fluid in an asymmetric channel [18], peristaltic flow of Herschel-Bulkley fluid [19] and peristaltic motion of Sutterby nanofluids [20]. Recently, many renewed studies for peristaltic flows of Newtonian and non-Newtonian fluids can be seen in [21,22,23,24,25,26] and references mentioned therein. All these reported studies are implemented numerical or analytically with deterministic methodologies while artificial intelligence (AI) algorithms based stochastic paradigm looks promising and capable alternative to be investigated/explored/exploited in the domain of peristaltic flows of Newtonian and non-Newtonian fluidic models.

The intelligent computing solvers are introduced exhaustively through AI algorithms by the researcher using neural networks trained with convex and non-convex optimization mechanisms for nonlinear standard integer as well as fractional order systems [27,28,29,30,31]. Recently intelligent computing solvers are implemented on different applications of paramount importance such as nonlinear circuit theory models [32,33], nonlinear singular fractional Lane-Emden systems [34], nonlinear optics [35], nonlinear Van-der Pol Mathieu’s oscillatory systems [36], nanotechnology [37,38], magnetohydrodynamics [39,40], nonlinear SITR model for novel COVID-19 dynamics [41], astrophysics [42], atomic physics [43], nonlinear singular boundary value problems [44], random matrix theory [45], electromagnetics [46,47], bioinformatics [48,49], financial models [50,51] and ordinary/partial fractional order differential equations [52,53,54]. Additionally, AI-based networks using Bayesian neural networks are exploited in different applications such as optimization of fluid flow processes [55,56], modeling of the explosion risk of the fixed offshore platforms [57], reliable optimization of complex equipment in automotive manufacturing [58], and solution dynamics of bio-convective nanofluidic models [59]. All this reported literature-inspired authors to examine the AI-based computing methodologies using Bayesian neural networks to solve the nonlinear differential systems governing the peristaltic flows of Newtonian and non-Newtonian fluidic models.

The aim of the presented communication is to design an intelligent Bayesian regularization backpropagation network (IBRBNs) based stochastic numerical computing framework to examine the dynamics of peristaltic motion of a third-grade fluid in a planar channel using multilayer structure modeling legacy of neural networks trained with efficient optimization strength of Bayesian regularization method. The reference datasets used as inputs and targets parameters of IBRBN have been obtained via the state-of-the-art Adams numerical method for multiple scenarios of peristaltic transport model by varying the volume flow rate, material parametric of a third-grade fluid model, wave amplitude, and inclination angles. The designed integrated IBRBNs are constructed by exploiting each epoch’s training, testing, and validation operations by introducing a figure of merit on mean square error sense. Comparative analysis on exhaustive numerical experimentation studies of presented computing platform IBRBNs with standard numerical solutions on mean square error, histograms and regression index substantiated the precision, stability, reliability, and robustness to solve a variant of peristaltic motion of a third-grade fluid in a planar channel.

The rest of the paper’s organization is outlined as follows: A compact description of problem formulation of peristaltic motion of a third-grade fluid is provided in Section 2. The designed scheme IBRBNs is briefly introduced in Section 3. Results of exhaustive simulation studies for IBRBNs with necessary interpretation are provided in Section 4. The concluded remarks are listed in the last Section 5.

2. Problem Formulation

The problem formulation for the peristaltic transport model of a third-grade incomprehensive fluid in a planner channel is briefly narrated here. The geometry of the problem is graphically illustrated in Figure 1.

We consider the problem of peristaltic slow of an incompressible third-grade fluid in the channel under long wavelength and low Reynolds number assumptions. The long wavelength and low Reynolds number are employed for simplification of the mathematical model with reduction of the dimensions of the peristaltic transport model The problem is governed by the following dimensionless ordinary differential equations as per the procedure adopted in reported studies [60,61,62]

$$\frac{\partial p}{\partial x}=\frac{{\partial }^3\psi }{\partial y^3}+2\beta \frac{\partial }{\partial y}{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^3=0$$

(1)

$$\frac{\partial p}{\partial y}=0,$$

(2)

where p is the dimensionless pressure, $\psi$ is the dimensionless stream function, $\beta$ is the dimensionless material parametric of the third-grade fluid model. (1) and (2) it turns out that $p\ne p\left(y\right)$ and therefore (1) becomes

$$\frac{{\partial }^4\psi }{\partial y^4}+2\beta \frac{{\partial }^2}{\partial y^2}{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^3=0.$$

(3)

Equation (3) subjected to the following boundary conditions

$$\begin{array}{l}\psi =0 \hspace{1em}\frac{{\partial }^2\psi }{\partial y^2}=0 \hspace{1em}at\hspace{1em}y=0\\ \psi =F\hspace{1em}\frac{\partial \psi }{\partial y}=-1\hspace{1em}at\hspace{1em}y=h=1+\varphi \sin x\end{array}$$

(4)

where F is the dimensionless volume flow rate in the wave frame, and h is given by $y=h=1+\varphi \sin x$ in which $\varphi \left(=\frac{b}{a}\right)$ is the ratio of the wave amplitude to the ratio of the channel.

Taking values, $\beta \ge 0; F=\left[-2,2\right]; \varphi =\left[0,1\right); x\in \left[0 to 2\pi \right] or x\in \left[-\pi to \pi \right]$ one may proceed to solve the Equation (3) as

$${\psi }^{\left(iv\right)}+2\beta \frac{{\partial }^2}{\partial y^2}{\left({\psi }^{″}\right)}^3=0$$

(5)

where prime represent the derivative with respect to ‘y’.

Simplifying Equation (5) as:

$${\psi }^{\left(iv\right)}+6\beta {\psi }^{\left(iv\right)}{{\psi }^{″}}^2+12\beta {{\psi }^{‴}}^2{\psi }^{″}=0$$

(6)

with boundary conditions as:

$$\begin{array}{l}\hspace{1em}\hspace{1em}\psi (0)=0, {\psi }^{″}(0)=0\\ \mathrm{for} y=h=1+\varphi \sin x\\ \hspace{1em}\hspace{1em}\psi (y)=F, {\psi }^{\prime }(y)=-1.\end{array}$$

(7)

3. Methodology with Performance Metrics

The methodology for solving the mathematical expression (6) and (7) representing the dynamics of peristaltic motion of a third-grade fluid in a planar channel model is provided in two steps. Firstly, the reference Adams numerical solutions for Equations (6) and (7) is calculated using ‘NDSolve’ routine with methods ‘Adams’ in Mathematica software environment for multiple scenarios of peristaltic transport model by varying the volume flow rate, material parametric of third-grade fluid model, wave amplitude and inclination angles. While in the second step, the developed solutions of each variation are used for the training, testing, and validation of IBBRNs. The developed or constructed settings of physical quantities in terms of cases and scenarios of the peristaltic transport systems are tabulated in

Table 1. Numerical values for the setting of each case and scenario of peristaltic motion model of a third-grade fluid.

Scenario	Case	β	θ	ϕ	F
1	1	0.08	π/2	0.4	−0.1
	2	0.07	π/2	0.4	−0.1
	3	0.04	π/2	0.4	−0.1
	4	0.02	π/2	0.4	−0.1
2	2	0.08	π/5	0.4	−0.1
	3	0.08	π/4	0.4	−0.1
	4	0.08	π	0.4	−0.1
3	2	0.08	π/2	0.1	−0.1
	3	0.08	π/2	0.6	−0.1
	4	0.08	π/2	0.7	−0.1
4	2	0.08	π/2	0.4	1.2
	3	0.08	π/2	0.4	1.4
	4	0.08	π/2	0.4	2.0

. The workflow diagram of IBBRNs is portrayed in Figure 2, in which the system model is defined in step 1, problem formulation with reference numerical solution is provided in step 2, the design methodology for the neural network is presented in step 3 while the solution results and analyses are given in step 4 and step 5, respectively.

3.1. Adams Numerical Scheme

The numerical solutions with Adams scheme [63] are determined for the peristaltic transport model in Equations (6) and (7) for each scenario as mentioned in Table 1. Adams numerical procedure is implemented by transforming the expression into a first-order equivalent system as briefly narrated below.

The general form of first-order differential system is given as follows:

$${\psi }^{\prime }=g(y,\psi ), \psi (y_0)={\psi }_0,$$

(8)

The first-order formulation of Equation (6) can be written as:

$$\begin{array}{l}{\psi }_1=\psi (y), {\psi }_2={{\psi }^{\prime }}_1={\psi }^{\prime }, {\psi }_3={{\psi }^{\prime }}_2={\psi }^{″}, {\psi }_4={{\psi }^{\prime }}_3={\psi }^{‴},\\ {\psi }_5=-6\beta {\psi }_5{\psi }_3^2-12\beta {\psi }_4^2{\psi }_3\end{array}$$

(9)

Then in the case of 2-steps predictor formulation, we have

$${\tilde{\psi }}_{n+1}={\psi }_n+1.5hg(y_n,{\psi }_n)-0.5hg(y_{n-1},{\psi }_{n-1}),$$

(10)

and accordingly, 2-steps corrector formulation, we have

$${\psi }_{n+1}={\psi }_n+0.5h(g(y_{n+1},{\tilde{\psi }}_{n+1})+g(y_n,{\psi }_n)),$$

(11)

On a similar pattern, higher step formulation for Adams methods can be given; however, the reference solutions are determined in the presented study by utilizing the adaptive procedure between steps 2 to 12 of the Adams method.

3.2. Bayesian Regularization Neural Networks

Bayesian regularization neural networks are considered to be more reliable, robust, and efficient than the standard backpropagation networks [55] and may diminish or eliminate the requirements of the cross-validation process in learning. Bayesian regularization procedure exploits a mathematical process that transformed a nonlinear regression problem into an equivalent well-posed statistical task by means of ridge regression. A more detailed description of Bayesian regularization can be seen in [55,64].

Normally, the Bayesian regularization is conducted within the Levenberg-Marquardt method, where the backpropagation is incorporated to determine the Jacobian ‘jX’ of the performance considering the weight and bias variables X. The adjustment of each variable follows the principle of the Levenberg-Marquardt algorithm as follows

$$\begin{array}{l}jj=jX\times jX,\\ je=jX\times E,\\ dX=\frac{-(jj+I\times mu)}{je},\end{array}$$

where E represents all errors while I denote the identity matrix. The adaptive controlling parameter mu is increased by the factor of mu_inc until the change reduced performance. The changes are then incorporated into the network, and parameter mu is decreased by factor mu_dec accordingly. The modification of the network is terminated for the attainment of any of the following predefined conditions

• The execution of maximum epochs.
• The maximum set time is exceeded.
• Performance goal is attained.
• The performance gradient is less than the minimum gradient value, i.e., min_grad.
• Adaptive parameter mu is greater than its maximum value, i.e., mu_max.

Theoretical proof of the convergence of Bayesian regularization networks is well established in the literature [55,60].

However, in the presented study the proposed IBBRNs are implemented using Matlab toolbox for neural networks with the help of the ‘nftool’ command. The developed solutions of numerical methods of each variation of the peristaltic transport model in Equations (6) and (7) are used arbitrarily for the training and testing samples of IBBRNs as input and target parameters. The implementation of neural networks via IBBRNs with the setting of a parameter such as training and testing data, layers structure, randomization of data, hidden neuron and log-sigmoid transfer function, etc. are provided in

Table 2. The parameter settings for implementation of IBBRNs.

Index	Settings
Maximum epochs for training	1000
Performance/fitness goal	0
Marquardt adjustment parameter, i.e., mu	0.005
Decreeing factor of mu, i.e., mu_dec	0.1
Increasing factor of mu i.e., mu_inc	10
Maximum of mu	1010
Maximum number of validation failures	Inf
Minimum value of performance gradient	10−7
Number of hidden neurons	35
Training and testing samples	80 and 20 percentage
Sample selection	Randomize
Input, output, and hidden layers	Single for all three
Reference data set generation	Adams method

.

The setting of parameters is conducted with extensive care, after exhaustive simulations, experience, and understanding of optimization behavior. A small perturbation of these parameters may lead to premature convergence of IBBRNs.

3.3. Performance Measuring Metrics

The optimization of the weight of proposed IBBRNs is performed with the Bayesian regularization method on the basis of different performance metrics as defined below:

Absolute error (AE), i.e., the absolute difference between the approximated $\widehat{\psi }(y)$ and reference $\psi (y)$ solutions, is mathematically defined as follows:

$$AE=\left|{\psi }_i-{\widehat{\psi }}_i,\right|$$

(12)

in case of $\psi (y_i)=ih\mathrm{for}i=1,2,\dots ,N$, where N represents the total number of grid points.

The mean square AE (MSE) is expressed as follows:

$$MSE=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left({\psi }_i-{\widehat{\psi }}_i\right)}^2}$$

(13)

While the metric of correlation R is defined as

$$R=1-\frac{{\displaystyle \sum _{i=1}^N\left({\widehat{\psi }}_i-{\psi }_i\right)}}{{\displaystyle \sum _{i=1}^N\left({\widehat{\psi }}_i-\overline{\psi }\right)}}.$$

(14)

here, $\overline{\psi }$ represents the average of target or reference value. Values of R equal to unity and zero MSE and AE represent perfect modeling of IBBRNs.

4. Results and Discussion

Results of numerical treatment with IBRBNs based stochastic numerical computing are presented here to examine the dynamics of peristaltic motion of a third-grade fluid in a planar channel. The proposed IBRBNs are implemented for 1, 2, 3, and 4 scenarios based on varying volume flow rate, material parametric of the third-grade fluid model, wave amplitude, and inclination angles, respectively. The numerical values of the parameters of the peristaltic motion model of a third-grade fluid are tabulated in Table 1 for each case of all four scenarios.

The mathematical relations of the peristaltic motion model (7) for cases 1 to 4 of scenario 1 are written as follows:

$${\psi }^{\left(iv\right)}+0.48{\psi }^{\left(iv\right)}{{\psi }^{″}}^2+0.96{{\psi }^{‴}}^2{\psi }^{″}=0,$$

(15)

$${\psi }^{\left(iv\right)}+0.42{\psi }^{\left(iv\right)}{{\psi }^{″}}^2+0.84{{\psi }^{‴}}^2{\psi }^{″}=0,$$

(16)

$${\psi }^{\left(iv\right)}+0.24{\psi }^{\left(iv\right)}{{\psi }^{″}}^2+0.48{{\psi }^{‴}}^2{\psi }^{″}=0,$$

(17)

$${\psi }^{\left(iv\right)}+0.12\beta {\psi }^{\left(iv\right)}{{\psi }^{″}}^2+0.24\beta {{\psi }^{‴}}^2{\psi }^{″}=0,$$

(18)

while the associated conditions are given as follows:

$$\psi (0)=0, {\psi }^{″}(0)=0, \psi (1)=-0.1, {\psi }^{\prime }(1)=-1,$$

(19)

and similarity relations for other cases of the peristaltic motion model are derived.

The reference solutions of the Adams method used as a dataset for IBBRNs are obtained for each variant of the peristaltic transport model as listed in Table 1 and are used in IBBRNs by the process of random segmentation into training and testing with 80% and 20%, respectively. The IBBRNs as structure in Figure 3 are used for finding the approximate solutions of the peristaltic transport model.

The results of the learning curves of IBBRNs, i.e., the iterative update of the MSE values again epochs are shown graphically in Figure 4 for scenario 1, Figure 5 for scenario 2 and 3, and Figure 6 for scenario 4 for each case of the peristaltic transport model. All these iterative learning curves show that the IBBRNs operate by attaining the average best performance values of MSE around 10⁻¹⁰ in almost every scenario of the peristaltic transport model.

The results of Bayesian regularization-based backpropagation algorithms are shown graphically in Figure 7 for scenario 1 for each case of the peristaltic transport model. One may see that gradient values attained around 10⁻⁸ and Marquardt adaptive parameter mu is equal to 500 for scenario 1 of peristaltic transport model and similar behavior of the results are observed for other three scenarios of peristaltic transport model. One may infer that the backpropagation algorithm of IBBRNs operates consistently and smoothly for each variation of the peristaltic transport model.

The goodness to fit the reference solutions for the proposed IBBRNs are shown graphically in Figure 8 for scenario 1, Figure 9 for scenario 2 and 3, and Figure 10 for scenario 4 for each case of the peristaltic transport model. The value of errors from reference solution for each sample in training and testing are also plotted in Figure 8, Figure 9 and Figure 10 One may observe that the results/outcomes of the proposed IBBRNs match with the standard solutions effectively, and error plots with negligible magnitude are further endorsed the reasonable precision for each case of all four scenarios of the peristaltic transport model.

Error histogram illustrations are used to analyze the precision of proposed IBBRNs, and results are shown graphically in Figure 11 for scenario 4 of the peristaltic transport model. A similar pattern of the results is observed for other scenarios of the peristaltic transport model. One may observe that the results/outcomes of the proposed IBBRNs are close to zero error bin consistently with the precision of order 10⁻⁵ for each case of all four scenarios of the peristaltic transport model.

Regression plots are used to further analysis of the performance, and the result of proposed IBBRNs are shown graphically in Figure 12, i.e., Figure 12a–d for scenarios 1, 2, 3, and 4, respectively of the peristaltic transport model. The value of the coefficient of determination R is consistently in close vicinity of the desired value of unity for almost all the cases of the peristaltic transport model. By taking the training and testing sample of 80% and 20% in IBBRNs, we obtained the results with MSE around 10⁻¹⁰ or below, therefore, we achieve the results of R = 1 in Figure 12. However, if we decrease the training sample percentage to around 50% or less then the overall values of the R lie around 0.7 or less due to more error in testing samples.

The numerical results of proposed IBBRNs are presented in

Table 3. Results of proposed IBBRNs for peristaltic transport model.

Scenario	Case	C		Performance Attained	Gradient Value	Mu Step Size	Epoch Executed	Consumed Time
		Training	Testing
1	1	2.00 × 10−13	1.03 × 10−9	2.20 × 10−13	9.28 × 10−8	5.00	148	0.00.06
	2	2.55 × 10−13	2.07 × 10−10	2.55 × 10−13	9.77 × 10−8	5.00	66	0.00.02
	3	4.34 × 10−13	5.37 × 10−12	4.34 × 10−13	1.40 × 10−8	5.00	13	0.00.01
	4	2.10 × 10−12	2.77 × 10−9	2.10 × 10−12	1.89 × 10−8	500	12	0.00.01
2	2	1.33 × 10−12	3.98 × 10−12	1.33 × 10−12	1.40 × 10−8	500	11	0.00.01
	3	1.59 × 10−12	2.50 × 10−10	1.59 × 10−12	3.82 × 10−8	500	12	0.00.01
	4	1.91 × 10−13	2.19 × 10−9	1.91 × 10−13	9.95 × 10−8	500	48	0.00.02
3	2	7.14 × 10−11	3.32 × 10−6	7.14 × 10−11	5.23 × 10−8	500	64	0.00.02
	3	9.90 × 10−13	1.65 × 10−11	9.89 × 10−13	9.80 × 10−8	500	22	0.00.01
	4	1.32 × 10−13	8.14 × 10−13	1.32 × 10−13	8.19 × 10−8	500	14	0.00.01
4	2	3.26 × 10−12	3.82 × 10−11	3.26 × 10−12	9.94 × 10−8	50	30	0.00.01
	3	1.06 × 10−11	1.05 × 10−10	1.06 × 10−11	9.84 × 10−8	50	32	0.00.01
	4	7.33 × 10−12	2.18 × 10−11	7.33 × 10−12	9.81 × 10−8	50	172	0.00.04

for MSE for both sets of data for training and testing samples along with the value of best performance for each case and scenario of the peristaltic transport model. The parameters of controlling the backpropagation procedure of Bayesian regularization methods are executed at each epoch with gradient and step size Mu index and their final results are also tabulated in Table 3. Additionally, the computational time-based complexity of IBBRNs for each case of the peristaltic transport model is also given in Table 3. One may see that the results of training are slightly better than that of testing inputs which are due to the fact that testing results are based on unbiased data. The performance on MSE is consistently in the close vicinity of 10⁻¹³. The time complexity is close to 5 s for almost every case of the peristaltic transport model, which shows the smooth operation of IBBRNs.

Accordingly, to analyze the effect of variation in physical parameters, the results of proposed IBBRNs are obtained for both solution and derivative for all four cases of four scenarios of the peristaltic transport model. These effects are presented in Figure 13, Figure 14, Figure 15 and Figure 16 for scenarios 1, 2, 3, and 4, of the peristaltic transport model, respectively. The IBBRNs consistency in accordance to reference numerical solution for each variation of peristaltic transport model. The absolute error-based analysis is presented in order to access the level of the precision, and results show the mapping of the order 10⁻⁴ to 10⁻⁶ for each case of all four scenarios of the peristaltic transport model.

5. Conclusions

The design of an intelligent Bayesian regularization backpropagation network-based stochastic numerical computing framework is presented to examine the dynamics of peristaltic motion of a third-grade fluid in a planar channel by using the multilayer structure modeling legacy of artificial neural networks learned with efficient optimization of Bayesian regularization based backpropagation methodology. The reference datasets used as inputs and targets parameters of IBRBN have been obtained via standard state of the art Adams numerical method for multiple scenarios of peristaltic transport model by varying the volume flow rate, material parametric of the third-grade fluid model, wave amplitude, and inclination angles. The designed integrated IBRBNs are constructed by exploiting the training (80%) and testing (20%) operations at each epoch by introducing a figure of merit on mean square error sense. Comparative analysis on exhaustive numerical experimentation studies of presented computing platform IBRBNs with standard numerical solutions on means square error (around 10⁻¹⁰ to 10⁻¹³), histograms (with maximum results close to reference zero error line), and regression index (consistently achieving close to the desired unity vale) substantiated the precision, stability, reliability, and robustness to solve a variant of peristaltic motion of a third-grade fluid in a planar channel.

In the future, one may conduct the investigation or exploration in the designed IBBRNs as an alternate, accurate and effective methodology for fractional processing based linear and nonlinear systems [65,66], nonlinear differential model arises in computer virus propagation fields [67,68] and nanofluidic models arise in mechanics [69,70].