Error Histogram Research Articles

Levenberg–Marquardt neural network-based intelligent computation for the non-Newtonian polymer during forward roll coating

Scientists and researchers widely recognize the effectiveness of artificial intelligence (AI)-based machine learning and intelligent computing solvers, demonstrating qualities such as resilience, robustness, stability, and rapid convergence. One particularly significant and rapidly growing field within AI is artificial neural networks. This research uses a supervised neural network model based on Levenberg–Marquardt backpropagation (LMB-SNNs) to examine the Sisko fluid model for the forward roll coating process (SFM-FRCP). A suitable transformation is applied to the partial differential equations based SFM-FRCP mathematical model, resulting in a set of nonlinear ordinary differential equations. The perturbation method has been used to find the analytical solutions for the velocity profile, pressure gradient, and pressure profile. A dataset for varying the pertinent parameters is generated, and the LMB-SNNs technique has been used to estimate the velocity profile, pressure gradient, and pressure profile behavior during FRCP for numerous scenarios. The numerical solution for SFM-FRCP in different scenarios, such as the validation, training, and testing procedures of LMB-SNNs, is carried out. Moreover, the state transition index, fitness outline, mean square error, histogram error, and regression presentation also endorse the strength and reliability of the solver LMB-SNNs for SFM-FRCP. The comparative analyses and performance studies through outputs of regression drawings, absolute error, and error histograms validate the effectiveness of the suggested solver LMB-SNNs. The method's precision is verified by the closest numerical outputs of both built and dataset values with similar levels 10−11–10−14. Furthermore, it has been observed that as the non-Newtonian parameter increases, the fluid velocity decreases. The research work carried out in this paper is original and fills a gap in the existing research by showing the rheological properties of the Sisko fluid model and the implementation of the LMB-SNNs during the FRCP.

Numerical and intelligent neuro-computational modelling with Fourier’s energy and Fick’s mass flux theory of 3D fluid flow through a stretchable surface

Current work explores the intricacies of magnetohydrodynamic and mix convectional boundary-layer flow concerning couple stress Casson nanofluid (CSCNF) dynamics via a 3D stretchable surface. The addition of active and passive control mechanisms for nanoscales creates an innovative dimension to the exploration. Remarkably, the analysis incorporates the influence of non-Fourier and non-Fickian heat and mass flux, alongside the effects of thermophoresis and Brownian diffusion, to systematically investigate the heat and mass transportation phenomena. The governing equations (PDEs) describing the MHD-3DCSCNF model are converted into a set of ordinary differential equations to facilitate the ANN analysis. Employing the bvp4c technique, a dataset is systematically generated for the back propagation artificial neural network with Levenberg–Marquardt Algorithm (BANN-LMA) through four distinct scenarios. Via accurate testing, validation, and training, the BANN-LMA produces estimated results for the MHD-3DCSCNF problem. The performance validation of BANN-LMA is executed through several metrics, involving the mean squared error, error histogram and regression analysis. The training process, characterized by minimizing the MSE through a gradient descent methodology with optimized weights, exhibits a compelling correlation R = 1, between the target and network output. Furthermore, the consistent convergence observed highlights the method robustness and reliability.

Dynamical analysis of nonlinear combined drug therapy model for HIV infection: Bayesian regularization technique intelligent networks

The study aims to provide an overview of the design and analysis of the differential system of the Nonlinear Combined Drug Therapy Mathematical (NCDTM) model for HIV Infection. This is achieved by utilizing the artificial Neural Network with Bayesian Regularisation technique intelligent network (ANNs-BRTIN), which is a stochastic method that enhances accuracy, reliability, and efficiency in the dynamics calculation process. The NCDTM model is defined along with experiments employing integer and nonlinear mathematical forms via four classes Where the number of uninfected cells in the blood and represented by A(t) the countable number of free virus particles in the blood is represented by S(t) when RTI therapy is administered denotes the number of infected cells in the Pre-RT phase with η dosage represented by I(t), D(t) represents when RTI therapy is administered using, reflects the number of infected cells in the post-RT phase η dosage. The numerical computing of the NCDTM system is performed with Adams methods, and these results are fed to the proposed ANNs-BRTINs for finding the approximated solution of four distinct instances by incorporating 15% of the data for testing and validation and 85% for training. The presented ANNs-BRTINs accuracy is illustrated by comparing the results from the acquired dataset of the Adam method for sundry scenarios. The use of regression for testing and residual, state transitions, mean square error, error histograms, and correlation in numerical replications of the ANNs-BRTINs is also explored to confirm their capacity, validity, consistency, correctness, and competence.

Inclined magnetic force impact on cross nanoliquid flowing with widening shallow and heat generating by using artificial neural network (ANN)

Artificial neural networks (ANNs) have a wide range of applications in science and technology. Artificial neural networks (ANNs) widely utilized in image and speech recognition, neural language processing, drug discovery, genomics and bioinformatics, Robotics and control systems, material science and energy and power system. Due to the above applications, the main focus of this article is to scrutinize the impact of inclined magnetic field on a cross nanofluid flow with slip velocity and convective boundary conditions over a variable porosity. Gold (Au) nanoparticles are suspended in base liquid blood. Thermal stratification and heat generation with joule heating impact is taken in the energy equation in order to see the heat transfer. To transform the non-linear partial differential equations (PDE's) into non-linear ordinary differential equations (ODE's) utilized the von-Karman similarity transformation parameters. For the solutions of the non-linear ODE's, the 4th- order Runge-Kutta numerical method is employed. A dataset for the employed neural network back propagated Levenberg-Marquard scheme (NN-BLMS) is generated for various estimations of the embedded parameters like Weissenberg number, magnetic parameter, angle of inclination, porosity and permeability parameters, thermal stratification parameter by utilizing the 4th-order Runge-Kutta numerical scheme. The testing, validation and training methods of NN-BLMS are utilized to scrutinize the approximate solution of cross nanofluid with slip velocity and convective boundary conditions. The performance of the proposed NN-BLMS to successfully solved the cross nanofluid model is endorsed via mean squared error, error histogram and regression analysis. Streamlines and heatlines are plotted for different parameters including in velocity and temperature equations.

Multilayer neural networks for studying three-dimensional flow of non-Newtonian fluid flow with the impact of magnetic dipole and gyrotactic microorganisms

The current paper explores the three-dimensional flow of an Oldroyd-B liquid with the impact of a magnetic dipole that occurred by stretching a flat surface placed in the plane with a linear velocity variation in two directions containing motile gyrotactic microorganisms. Using proper similarity transformations, the governing equations are reduced into nonlinear coupled ordinary differential equations (ODEs). The ODEs are then solved using Runge–Kutta-Fehlberg (RKF) method. The training, testing, and validation processes are carried out in parallel to adapt neural networks and calculate an approximate solution for the considered model. This helps to reduce the mean square error (MSE) function by Levenberg–Marquardt backpropagation. The efficiency of the suggested backpropagated neural networks methodology has been demonstrated by utilizing outcomes such as MSE, error histograms, correlation and regression. Results reveal that the heat transport augments for increased Biot number values. The mass transport declines for improved chemical reaction rate parameter values. A higher Peclet number will result in a lower motile diffusivity and result in a decline in the micro-organism’s density profile. For the least value of Mu and gradient, better convergence of the findings can be achieved with better network testing and training.

A hybrid machine learning algorithm for studying magnetized nanofluid flow containing gyrotactic microorganisms via a vertically inclined stretching surface.

The novelty of the present work is to acquire continuous functions as solutions rather than the discrete ones that traditional numerical methods generally produce and to minimize simulation times and higher computation costs that are the fundamental barriers to employing any numerical method. In this study, a novel hybrid finite element-based machine learning algorithm utilizing the Levenberg-Marquardt scheme with backpropagation in a neural network (LMBNN) is presented to analyze the nanofluid flow in the presence of magnetohydrodynamics and gyrotactic microorganisms through a vertically inclined stretching surface in a porous medium. Finite Element Method is used to generate the minimum reference dataset for LMBNN by varying six flow parameters in the form of six scenarios. Surface plots are utilized to understand how these scenarios affect velocity, temperature, concentration of nanoparticles, and density of motile microorganisms. Regression analysis, error histogram analysis, and fitness curves based on mean square error all support the LMBNN's effectiveness and dependability. Results reveal that temperature increases with the rise in Brownian motion and thermophoresis parameter, whereas the reverse trend has been noticed for Prandtl number. The motile microorganism density number decreases with the rise in Prandtl numbers but improves with the porosity parameter.

A novel radial basis procedure for the SIRC epidemic delay differential model

The purpose of this work is to construct a reliable stochastic framework for solving the SIRC delay differential epidemic system, i.e. SIRC-DDES that is based on the coronavirus dynamics. The design of radial basis (RB) transfer function with the optimization of Bayesian regularization neural network (RB-BRNN) is presented to solve the SIRC-DDES. The SIRC-DDES is classified into susceptible , infected , recovered and cross-immune . The exactness of the RB-BRNN is performed for three cases of SIRC-DDES by using the performances of the obtained and reference results. The mean square error is reduced by using the training, testing and substantiation performances with the reference solutions. The small values of the absolute error around 10−07 to 10−08 and different statistical operator performances based on the error histogram values, transitions of state investigations, correlation and regression tests also approve the accuracy of the proposed technique.

A computational stochastic procedure for solving the epidemic breathing transmission system

This work provides numerical simulations of the nonlinear breathing transmission epidemic system using the proposed stochastic scale conjugate gradient neural networks (SCGGNNs) procedure. The mathematical model categorizes the breathing transmission epidemic model into four dynamics based on a nonlinear stiff ordinary differential system: susceptible, exposed, infected, and recovered. Three different cases of the model are taken and numerically presented by applying the stochastic SCGGNNs. An activation function ‘log-sigmoid’ uses twenty neurons in the hidden layers. The precision of SCGGNNs is obtained by comparing the proposed and database solutions. While the negligible absolute error is performed around 10–06 to 10–07, it enhances the accuracy of the scheme. The obtained results of the breathing transmission epidemic system have been provided using the training, verification, and testing procedures to reduce the mean square error. Moreover, the exactness and capability of the stochastic SCGGNNs are approved through error histograms, regression values, correlation tests, and state transitions.

Novel intelligent Bayesian computing networks for predictive solutions of nonlinear multi-delayed tumor oncolytic virotherapy systems

Oncolytic viral immunotherapy is gaining considerable prominence in the realm of chronic diseases treatment and rehabilitation. Oncolytic viral therapy is an intriguing therapeutic approach due to its low toxicity and dual function of immune stimulations. This work aims to design a soft computing approach using stupendous knacks of neural networks (NNs) optimized with Bayesian regularization (BR), i.e. NNs-BR, procedure. The constructed NNs-BR technique is exploited in order to determine the approximate numerical treatment of the nonlinear multi-delayed tumor virotherapy (TVT) models in terms of the dynamic interactions between the tumor cells free of viruses, tumor cells infected by viruses, viruses, and cytotoxic T-lymphocytes (CTLs). The strength of state-of-the-art numerical approach is incorporated to develop the reference dataset for the variation in the infection rate for tumor cells, virus-free tumor cell clearance rate by CTLs, CTLs clearance rate for infectious tumor cells, the natural lifecycle of infectious tumor cells, the natural lifecycle of viral cell, the natural lifecycle of CTLs cells, tumor cells free of viruses’ maximum proliferation rate, production of tumor cells with an infection, CTLs simulated ratio for infectious tumor cells, CTLs simulated ratios for virus-free cells and delay in time. The dataset is randomly chosen/segmented for training-testing-validation samples to construct the NNs models optimized with backpropagated BR representing the approximate numerical solutions of the dynamic interactions in the TVT model. The performance of the designed NNs-BR technique is accessed/evaluated and outcomes are found in good agreement with the reference solutions having the range of accuracy from 10[Formula: see text] to 10[Formula: see text]. The efficacy of NNs-BR paradigm is further substantiated after rigorous analysis on regression metrics, learning curves on MSE, and error histograms for the dynamics of TVT model.

Design and analysis of cyber warfare model with intelligent predictive stochastic networks for attack–defend strategies on critical infrastructures

The purpose of this study is to develop a novel multi-class differential system (target network, protected network, inside attacker, and outside attacker) for a cyber war model that imitating the cyber war scenarios on different attack surfaces in the presence of multifarious attack vectors and analysis has been presented by stupendous knacks of intelligent predictive stochastic networks (IPSNs). The theoretical convergence analysis through attack-free and attack-invasion equilibrium in terms of basic reproduction number R0 has been presented along with the global stability of attack-free equilibrium points using the Lyapunov function. The reference solutions for studying the dynamics of all four classes of cyber war models in an air-gapped environment have also been determined in this paper for various scenarios by varying the protection strategy parameter ρ, the recovery strategy parameter ɛ, and the rate of attack from an outside/inside attacker using the Adams numerical method. The obtained data has been used in IPSNs as input/targets for training, validation, and testing in order to compute the approximate solution of the model. Extensive simulations have been run to determine the accuracy of the proposed framework for a complex advanced persistent attack targeting industrial systems. The outcomes of IPSNs have shown a fine resemblance with the matching of order 10−7 from the observed data. Also, the analysis based on mean square error convergence, error histogram, and regression index has further verified the strength and worth of the presented strategy.

On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks

Abstract The aim of this work is to solve a mathematical model based on the migration and emigration effects. The designed mathematical model shows one of the forms of prey-predator. The migration factor represents a step function for both normal and individuals that is restrictions or movement of people. The numerical solutions of the designed model are presented using the stochastic computational schemes based on the artificial neural networks (ANNs) together with the Levenberg-Marquardt back propagation (LMB), i.e., ANNs-LMB for solving the model based on the migration and emigration effects. Three different cases have been performed to solve the model based on the migration and emigration effects with the ANNs-LMB solver in terms of authentication, training, sample statistics and testing. The selection of the data is chosen as 80%, 10%, 10% for training, testing and authentication, respectively. The numerical results through the ANNs-LMB of the model based on the migration and emigration effects will be compared with the Runge-Kutta method. The results of the model based on the migration and emigration effects using the ANNs-LMB are provided to reduce the mean square error (MSE). For the capability and efficiency of the proposed ANNs-LMB, the numerical results are provided using the correlation, error histograms, regression and MSE.

On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks

Abstract The aim of this work is to solve a mathematical model based on the migration and emigration effects. The designed mathematical model shows one of the forms of prey-predator. The migration factor represents a step function for both normal and individuals that is restrictions or movement of the people. The numerical solutions of the designed model are presented using the stochastic computational schemes based on the artificial neural networks (ANNs) together with the Levenberg-Marquardt back propagation (LMB), i.e., ANNs-LMB for solving the model based on the migration and emigration effects. Three different cases have been performed to solve the model based on the migration and emigration effects with the ANNs-LMB solver in terms of authentication, training, sample statistics and testing. The selection of the data is chosen as 80%, 10%, 10% for training, testing and authentication, respectively. The numerical obtained results through the ANNs-LMB of the model based on the migration and emigration effects will be compared with the Runge-Kutta method. The results of the model based on the migration and emigration effects using the ANNs-LMB are provided to reduce the mean square error (MSE). For the capability and efficiency of the proposed ANNs-LMB, the numerical results are provided using the correlation, error histograms, regression and MSE.

Neuro-computing for third-grade nanomaterial flow under impacts of activation energy and mixed convection along rotating disk

This paper examines the activation energy influence in third-grade nanoparticle flow model (TG-NPFM), which is nonlinear mixed convective flow over a spinning disk under the influence of heat sink/source as well as viscous dissipation by utilizing Bayesian Regulation Method with backpropagated Artificial Neural Networks (BRM-BPANN). Nonlinear thermal radiation is also involved in the considered flow dynamics to obtain the approximated numerical solutions. The nonlinear PDEs of TG-NPFM are then transformed into nonlinear ODEs by implementing the corresponding transformation. We solved these ODEs by Optimal Homotopy Analysis Method (OHAM) to explain the dataset used as a reference for BRM-BPANN for different scenarios of TG-NPFM. This reference dataset is then exported to MATLAB to compute the results. The outcomes of TG-NPFM are figured by adopting the procedures of testing, validation and training. Moreover, approximated solution is compared with standard solution and the efficacy examination of TG-NPFM is authenticated by the studies of MSE, error histogram and regression plots. These soft computation frameworks provide incentive to use an efficient and dependable alternative paradigm built on soft computing environments to solve problems by doing a descriptive analysis to mitigate the impacts of different physical features. It is a new implementation of intelligent computational system of artificial intelligence introduced by incorporating the solver BRM-BPANN for interpreting the TG-NPFM. The absolute error values lie between 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text], 10[Formula: see text] to 10[Formula: see text] and 10[Formula: see text] to 10[Formula: see text], which show the reliability and accuracy of the technique. The convergence and precision of the algorithm can easily be seen through the results of performance, training state and fitness plot, along with the regression value of [Formula: see text].

Determination of the code reliability of digital angle converters

To measure the angular motion parameters of various kinds of objects, digital angle converters are widely used, which differ in the output electrical interface using a parallel or serial method of code transmission. Currently, there is no single approach to the metrological assessment of the accuracy characteristics of these digital angle converters. Such an integral assessment can be the code reliability of the converter, which characterizes the probability that the value of the measured angle corresponds to the calculated quantum, the code of which is read. Now there is an analytical method for calculating the code reliability used for digital angle converters with parallel code transmission. The article proposes an algebraic method for calculating the code reliability. Unlike the analytical method, it lacks the disadvantages of approximating the error histogram of the digital angle converters by random variable distribution functions. The conducted studies have shown that the algebraic method of calculating the code reliability can be used for the digital angle converters, with parallel and serial code transmission. It is shown that in order to fi nd the unevenness of the angular scale of the digital angle converters using a dynamic goniometer with an incremental angle sensor, it is necessary to center the array of measured error values. The results obtained can be used in measuring technology and metrology for digital angle converters calibration. They are of interest to specialists working in the fi eld of metrology of angular measurements, measuring equipment, digital angle converters manufacturers.

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

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 (β), magnetic field (M), porosity (S), nanoparticle concentrations (ϕ1=ϕ2=ϕ3), 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 β grows, whereas velocity increase as M increases. The concentrations of nanoparticles and thermal radiations have increasing effects on θ0. 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.

A radial basis deep neural network process using the Bayesian regularization optimization for the monkeypox transmission model

The motive of this work is to provide the numerical performances of the monkeypox transmission mathematical model by using a novel deep neural network process with eleven and twenty-two neurons in the hidden layers. The purpose to provide the deep neural network stochastic process is to obtain more accurate solutions of the monkeypox transmission mathematical system. This process is enhanced by using an activation radial basis function in both layers for solving the monkeypox transmission mathematical model along with the implementation of the Bayesian regularization optimization scheme. The presentation of the mathematical dynamical model has two categories, human and rodent. The human dynamics is classified into, susceptible, exposed, infectious, clinically ill human and recovered individuals. The rodent is divided into three forms, susceptible, exposed, and infected. A dataset is presented with the Adam approach that is processed using the training, testing, and certification procedure by taking the data as 0.13, 0.12 and 0.15. The correctness is observed through the matching of the results and the statistical plots are plotted using the regression, state transition, error histograms and correlation.

Thermal Enhancement in the Ternary Hybrid Nanofluid (SiO2+Cu+MoS2/H2O) Symmetric Flow Past a Nonlinear Stretching Surface: A Hybrid Cuckoo Search-Based Artificial Neural Network Approach

In this article, we considered a 3D symmetric flow of a ternary hybrid nanofluid flow (THNF) past a nonlinear stretching surface. The effect of the thermal radiation is considered. The THNF nanofluid SiO2+Cu+MoS2/H2O is considered in this work, where the shapes of the particles are assumed as blade, flatlet, and cylindrical. The problem is formulated into a mathematical model. The modeled equations are then reduced into a simpler form with the help of suitable transformations. The modeled problem is then tackled with a new machine learning approach known as a hybrid cuckoo search-based artificial neural network (HCS-ANN). The results are presented in the form of figures and tables for various parameters. The impact of the volume fraction coefficients ϕ1, ϕ2, and ϕ3, and the radiation parameter is displayed through graphs and tables. The higher numbers of the radiation parameter (Rd) and the cylinder-shaped nanoparticles, ϕ3, enhance the thermal profile. In each case, the residual error, error histogram, and fitness function for the optimization problem are presented. The results of the HCS-ANN are validated through mean square error and statistical graphs in the last section, where the accuracy of our implemented technique is proved.

Analysis of non-linear RIM system and neural computing of ringworm spread using the Levenberg–Marquardt back propagated scheme: Supervised learning

In this article application of neural network using Levenberg–Marquardt Back-propagation is implemented on differential model to study and analyze ringworm infectious disease. The formulated system of differential equations is consisting of following parts, namely; S(t) population which is at verge of being infected by ringworm, E(t) shows the environment effected by dermetophytosis fungus, I(t) whereas represent the infected individuals, R(t) and shows the population which has recovered from the infection. The solutions of different categories are represented by considering distinct datasets modeled and designed using LMB neural network. The numerical scheme Adam has been employed to establish a reference data set of the designed LMB neural network. The approximate outcomes of the SEIR based on dispersing and curing are discussed using the authentication, testing and training procedures to truncate the mean square error in function with help of LMB. The mean square error, regression, error histograms are generated to produce efficiency, effectiveness and correctness of proposed LMB neural network scheme.

Artificial intelligence computing analysis of fractional order COVID-19 epidemic model

Artificial intelligence plays a very prominent role in many fields, and of late, this term has been gaining much more popularity due to recent advances in machine learning. Machine learning is a sphere of artificial intelligence where machines are responsible for doing daily chores and are believed to be more intelligent than humans. Furthermore, artificial intelligence is significant in behavioral, social, physical, and biological engineering, biomathematical sciences, and many more disciplines. Fractional-order modeling of a real-world problem is a powerful tool for understanding the dynamics of the problem. In this study, an investigation into a fractional-order epidemic model of the novel coronavirus (COVID-19) is presented using intelligent computing through Bayesian-regularization backpropagation networks (BRBFNs). The designed BRBFNs are exploited to predict the transmission dynamics of COVID-19 disease by taking the dataset from a fractional numerical method based on the Grünwald–Letnikov backward finite difference. The datasets for the fractional-order mathematical model of COVID-19 for Wuhan and Karachi metropolitan cities are trained with BRBFNs for biased and unbiased input and target values. The proposed technique (BRBFNs) is implemented to estimate the integer and fractional-order COVID-19 spread dynamics. Its reliability, effectiveness, and validation are verified through consistently achieved accuracy metrics that depend on error histograms, regression studies, and mean squared error.

Visual Anomaly Detection via Partition Memory Bank Module and Error Estimation

Reconstruction method based on the memory module for visual anomaly detection attempts to narrow the reconstruction error for normal samples while enlarging it for anomalous samples. Unfortunately, the existing memory module is not fully applicable to the anomaly detection task, and the reconstruction error of the anomaly samples remains small. Towards this end, this work proposes a new unsupervised visual anomaly detection method to jointly learn effective normal features and eliminate unfavorable reconstruction errors. Specifically, a novel Partition Memory Bank (PMB) module is proposed to effectively learn and store detailed features with semantic integrity of normal samples. It develops a new partition mechanism and a unique query generation method to preserve the context information and then improves the learning ability of the memory module. The proposed PMB and the skip connection are alternatively explored to make the reconstruction of abnormal samples worse. To obtain more precise anomaly localization results and solve the problem of cumulative reconstruction error, a novel Histogram Error Estimation module is proposed to adaptively eliminate the unfavorable errors by the histogram of the difference image. It improves the anomaly localization performance without increasing the cost. To evaluate the effectiveness of the proposed method for anomaly detection and localization, extensive experiments are conducted on three widely-used anomaly detection datasets. The encouraging performance of the proposed method compared to the recent approaches based on the memory module demonstrates its superiority.