heat and mass transfer. This paper proposes a dynamic control model based on adaptive-network-based fuzzy

inference system (ANFIS), weighted logistic regression and robust relevance vector machine (RRVM).

Suitable similarity variables are applied to convert the flow equations into higher order ordinary differential

equations and solved numerically. The surface-contour plots are utilized to visualize the influence of active

parameters on velocity, thermal, nanoparticles concentration and motile microorganism’s density. The hybrid-

learning algorithm comprised of gradient descent and least-squares method is employed for training the ANFIS.

A novel RRVM is presented to predict the endpoint. RRVM solves the problem of sensitivity to outlier

characteristic of classical relevance vector machine (RVM), thus obtaining higher prediction accuracy. The key

idea of the proposed RRVM is to introduce individual noise variance coefficient to each training sample. In the

process of training, the noise variance coefficients of outliers gradually decrease so as to reduce the impact of

Key-Words:- ANFIS, Heat and Mass Transfer, MHD flow, Monte Carlo Simulation, Outliers, Robust

Classification, Robust RVM, Sparsity, Weighted Logistic Regression.

Received: June 25, 2021. Revised: January 15, 2022. Accepted: February 26, 2022. Published: March 30, 2022.

The magnetohydrodynamic (MHD) heat and mass

transfer processes over a moving surface are of

interest engineering and geophysical applications

such as geothermal reservoirs, thermal insulation,

are mostly determined by the cooling liquid

employed and the velocity of stretching. Some

polymer fluids with higher electromagnetic

fluid flows that are incompressible and viscous near

a porous infinite plate with arbitrary time dependent

heating plate. Senapati et al. [3] published the

results of chemical reactions of viscous fluids that

are electrically conducting via a porous material in

two-dimensional steady free convection flow along

a vertical surface with slip flow area.

properties play an important role in the

intensification of mixing processes in a variety of

has gotten a lot of attention. Several researches on

the heat and mass transfer properties of magneto

Carreau nanofluids with diverse characteristics such

as heat source/sink, thermal radiation,

suction/injection, and changing thermal conductivity

over a permeable/impermeable stretched sheet have

been conducted, see [4,5]. Nanofluids improve heat

transmission and can be used to improve the

efficiency of heat exchangers and reactors. In

nanofluids, bioconvection improves mass transfer,

output data via hybrid learning. This inference

describes numerous nodes in each tier. Fixed nodes,

shown by circles, represent parameter sets that are

fixed in the system, whereas adaptive nodes,

denoted by squares, represent parameter sets that are

modifiable in these nodes. The current layer's input

will be the output data from the preceding levels'

nodes.

proposed the relevance vector machine (RVM). The

RVM is a Bayesian evidence-based nonlinear

approach, often known as the "evidence process."

For each of the model coefficients, an independent

zero-mean Gaussian prior is assumed, as well as an

independent Gamma hyper prior for each

hyperparameter. After that, a training data set is

used to determine posterior distributions of model

coefficients and hyperparameters. Initially, the

posterior distributions were calculated using the

type-II maximum likelihood approach, which is an

evidence procedure. A variational inference

plate in oscillatory velocity suction. Noor et al. [11]

flow on an inclined surface with heat source/sink

effects. MHD fluid flow across a spinning disc was

researched by Turkyilmazoglu [12]. He analysed the

viscous dissipation and Joule heating components

using a spectral numerical integration approach.

Chen [13] studied heat and mass transport in MHD

free convective flow with Ohmic heating and

viscous dissipation using a numerical technique.

been rapidly developed. It is based on the notion of

improves processing capabilities by translating data

into a high-dimensional space and employing kernel

functions. In addition, Müller et al. [15] established

model of MHD heat and mass transport processes is

created. In order to achieve the intended control

effect, the RRVM model must be well-trained as an

identifier to approximate the link between input and

output and correctly anticipate the endpoint carbon

RRVM obtained by RBF kernel with.

(TRVM) was suggested by Yuan et al. [17], which

redefined the likelihood function as a trimmed one.

During model training, outliers are removed, and a

weighted technique is used to determine the

trimmed subset. The new technique can detect

outliers and improve the model's robustness. Many

robust methods are discussed by many papers in

several models, see e.g. [23-26].

on how the MHD heat and mass transfer

adaptation to convective effects is also

The problem boundary conditions are defined as

󰇛󰇜

 as  (6)

where 󰇛󰇜 denotes the components of velocity

along 󰇛󰇜directions, t is the time, 

represent the fluid temperature, nanoparticles

volume fraction and motile micro-organisms

of base fluid,  indicate the Brownian,

thermophoretic and microorganism diffusion

and maximum cell swimming speed ( -

constant),  signify the velocity,

thermal, concentration and microorganism slip

rules. As for the  fuzzy rule, the degree of



weighting strategy can be employed to the standard

where  is a weight associated with the

outlying observation, the effect of an outlier can be

reduced and therefore a robust decision boundary

can be obtained. Then, one question is raised: how

the concept of a weighted loss can be transformed

into the maximum likelihood approach. From Eq.

(19), the following relationship can be obtained:

dealt with in the maximum likelihood approach by

considering a complex model, the regularization

concept has been used in machine learning. By

employing the regularization concept to the original

where  is a regularization parameter which

controls the smoothness of a decision boundary and

󰇛󰇜 denotes a regularization term which

represents a penalty for a complex decision

boundary.

scalar-measured output. Assume that the objectives

with extra noise  as follows:

The posterior distribution over  is also Gaussian

and can be expressed as:

where  is the mean value vector and  is the

variance matrix, which are expressed as formulas

(29) and (30), respectively:

where 󰇛󰇜. The posterior

distribution over  are determined by

hyperparameters and , thus the hyperparameters

 denotes the  th diagonal element of matrix

. In the process of training,

Equations (13) – (16) are calculated iteratively.

Most of  tend to ward infinity and the

corresponding  will tend toward zero. The training

stops until all the hyperparameters are convergent or

the maximum number of iterations is reached.

each training samples noise  is a mean-zero