Abstract. By combining the linear canonical transform and the Riesz transform, we introduce the linear canonical Riesz transform (LCRT), which is further proved to be a linear canonical multiplier. Using this LCRT multiplier, we conduct numerical simulations on images. Notably, due to the fact that the linear canonical transform itself admits a fast algorithm, the LCRT can achieve a computational speed comparable to that of the linear canonical transform. Based on this, we introduce the new concept of the sharpness [Formula: see text] of the edge strength and continuity of images associated with the LCRT and, using it, we propose a new LCRT image edge detection method (LCRT-IED method) and provide its mathematical foundation. Our experiments indicate that this sharpness [Formula: see text] characterizes the macroscopic trend of edge variations of the image under consideration, while this new LCRT-IED method not only controls the overall edge strength and continuity of the image, but also excels in feature extraction in some local regions. These highlight the fundamental differences between the LCRT and the Riesz transform, which are precisely due to the multiparameter of the former. This new LCRT-IED method might be of significant importance for image feature extraction, image matching, and image refinement.

This article introduces the Bayesian probabilistic fuzzy neural network (BPFNN), a unified architecture designed to overcome the challenges of conventional fuzzy clustering and neural networks in terms of uncertainty, noise, and interpretability. At its core, the Bayesian probabilistic fuzzy $C$ -means (BPFCMs) algorithm is employed to define the hidden-layer nodes, extending traditional FCM through non-Gaussian modeling and posterior inference via Markov chain Monte Carlo (MCMC). By combining Metropolis-Hastings (MHs) for membership updates with Gibbs sampling for parameter estimation, BPFCM yields probabilistic memberships that capture uncertainty in the antecedent rules more effectively than deterministic approaches. Since the hidden-layer activations represent only similarity values between inputs and cluster centers, the original input features are not directly preserved. To compensate, the hidden-to-output connections are formulated as linear functions of the input, ensuring recovery of discriminative information in the consequent rules. These functions are optimized using a generalized cross-entropy (GCE) objective, with iteratively reweighted least squares (IRLSs) employed for efficient and regularized updates. Extensive experiments on benchmark datasets and high-dimensional laser-induced breakdown spectroscopy (LIBS) spectral data confirm that BPFNN consistently surpasses both classical fuzzy systems and contemporary deep learning models, providing improved accuracy, robustness, and interpretability.

A novel reinforced dual-flow neural network based on attention and a polynomial-based radial basis function network (DFBTP) is proposed to enhance classification performance on tabular data. DFBTP consists of two types of architectures such as advanced Transformer model based on Tabular Prior data Fitted Network (TabPFN) and SV-clustering driven radial basis function neural network (SV-PRBFNN). The conventional PRBFNN model may encounter local minima and noise issues during training, which negatively impacts its performance. Local minima result from initial parameter choices, and noise issues are inherent in the dataset. By introducing the Transformer model and the Whale Optimization Algorithm (WOA), these challenges can be mitigated. Within the dual-flow architecture, the attention flow is trained using Bayesian inference capabilities and structural causal models, and it uses the self-attention mechanism to capture global features. This approach mitigates the problem of local minima. SV-PRBFNN flow uses the fuzzy clustering algorithm based on support vectors to replace the original radial basis function for training. Fuzzy clustering based on support vectors can alleviate the negative impact of outliers on model performance and also reduce the number of fuzzy rules. During neural network hyperparameter optimization, WOA is used to identify the global optimal values for hyperparameters. In the experiments, DFBTP demonstrated its superiority in classification accuracy in comparative experiments on 18 datasets and 13 models, and also performed well on real-world datasets. The robust performance of DFBTP was further validated through statistical analysis of the experimental results.

Polynomial neural networks (PNN) have emerged as an effective regression modeling methodology in computational intelligence, relying on its interpretable polynomial nodes to fit complex nonlinear data relationships and the adaptive nature of self-organizing networks. To break the bottleneck of PNN structure in the field of multi-classification, this study designs a dynamical multiple polynomial-based neural networks (DMPNN) classifier, focusing on developing a flexible polynomial network classification methodology that enhances predictive capabilities without sacrificing the advantages of PNN structures. Our approach effectively addresses the challenges of multi-class classification with uncertain class boundaries and reduces computational complexity, which is achieved through the synergy of several proposed techniques. Three key issues underpin the proposed DMPNN: (a) The integration of PNN regression models using the one-against-all strategy can provide effective and scalable solutions to multi-class classification problems, especially for uncertain class boundary issues. (b) The dual statistical selection (DSS) approach aims to eliminate redundant inputs during data processing, reduce the computational burden, and increase the variety of neural network nodes in the model neuron selection stage. (c) The synergy of regularization methods including the ℓ2 norm-based method (ℓ2-LSM) and the DropFilter, is exploited to mitigate potential overfitting in coefficient estimation and enhance the generalization capabilities of the proposed classifier. A series of ablation experiments and parameter analysis were conducted to demonstrate the stability and reliability of the proposed model. Then, we applied DMPNN to 17 publicly available datasets and two engineering applications: Recycling of black plastic wastes and phased resolved partial discharge. The performance results show that the DMPNN model outperforms five classical classifiers and four state-of-the-art (SOTA) classifiers on 78.94% of the datasets. This highlights the unique ability of the proposed DMPNN to enhance predictive accuracy while maintaining model simplicity and interpretability.

In this study, we propose a novel data transformation-driven fuzzy clustering neural network (DTFCNN) to enhance the dimensionality reduction function and end-to-end refinment learning ability of the entire structure. Unlike conventional fuzzy clustering-based neural networks, which rely on fuzzy c-means clustering in the hidden layer and least squares error-based learning for connection weights, the DTFCNN utilizes backpropagation (BP) learning as a refinement algorithm. This refinement process enables iterative fine-tuning of the model’s parameters, allowing it to adapt more effectively to complex patterns. By using BP, DTFCNN enhances its ability to learn intricate feature interactions and extract relevant features from high-dimensional spaces, thereby significantly improving the model’s flexibility and performance. The proposed DTFCNN consists of four layers. First, a preprocessing layer employs principal component analysis for feature extraction and dimensionality reduction, where eigenvectors are used as connection weights for the preprocessing layer. Second, a hidden layer utilizes fuzzy c-means clustering for generating fuzzy membership degrees, and centers serve as connection weights for hidden layers. The entries of the partition matrix also are regarded as membership degrees. In the output layer, a linear-driven affine transform is used for fitting connection weights, and a SoftMax function is employed to express the outputs as probabilities. Finally, all parameters, such as eigenvectors, centers, and coefficients, from the preprocessing layer to the output layer are refined through BP-based learning. To validate the effectiveness of the DTFCNN, we conducted a collection of comparison experiments: 1) publicly benchmark datasets with statistical analysis, 2) facial recognition datasets for application-specific testing, and 3) three large-scale datasets. The results demonstrate that the DTFCNN outperforms classical classifiers, state-of-the-art fuzzy classifiers, and deep learning baselines in terms of accuracy and generalization capability. The DTFCNN model uses fewer parameters than deep learning baselines, resulting in faster training times without compromising performance. Overall, the DTFCNN achieves higher accuracy while maintaining model adaptability.

Original Article: Robust Classification via Interval Type-2 Fuzzy C-Means and Gradient Boosting, IEEE Transactions on Fuzzy Systems, vol. 33, no. 9, pp. 3103–3117, Sept. 2025. doi:10.1109/TFUZZ.2025.3583051.

This article introduces the Bayesian probabilistic fuzzy neural network (BPFNN), designed to overcome the limitations of Fuzzy C-Means (FCM) clustering, which struggles with uncertainty, noise, nonlinearity, and interpretability. Additionally, it addresses the shortcomings of traditional objective functions, such as mean squared error (MSE), which fail to capture the complexities inherent in high-dimensional and uncertain datasets. The BPFNN framework integrates Bayesian probabilistic modeling with advanced fuzzy clustering techniques, utilizing a non-Gaussian probability density function to better represent data uncertainties. A hybrid Markov chain Monte Carlo strategy, combining Metropolis-Hastings for membership updates and Gibbs sampling for cluster parameter estimation, is employed to effectively model uncertainty. For the learning of connection weights, the generalized cross-entropy loss function is applied, and the iteratively reweighted least squares algorithm is used to update the weights, allowing for a more precise quantification of the divergence between predicted and ground truth labels. Experimental evaluations on several benchmark datasets, as well as a high-dimensional laser-induced breakdown spectroscopy (LIBS) spectral dataset, demonstrate that the proposed BPFNN significantly outperforms both traditional methods and State-of-the-Art techniques in terms of classification accuracy and robustness. Notably, BPFNN achieves an average accuracy improvement of 3.2% over conventional models on benchmark datasets, with a 5.3% improvement on the LIBS dataset, highlighting its substantial advancement in the field.

Abstract. Via chirp functions from fractional Fourier transforms, we introduce fractional Riesz potentials related to chirp functions, which are further used to give a new image encryption method with double phase coding. In a comparison with the image encryption method based on fractional Fourier transforms, via a series of image encryption and decryption experiments, we demonstrate that the symbols of fractional Riesz potentials related to chirp functions and the order of fractional Fourier transforms essentially provide greater flexibility and information security. We also establish the relations of fractional Riesz potentials related to chirp functions with fractional Fourier transforms, fractional Laplace operators, and fractional Riesz transforms, and we obtain their boundedness on rotation invariant spaces.

Fuzzy clustering-based neural networks (FCNNs) based on information granulation techniques have been shown to be effective Takagi–Sugeno (TS)-type fuzzy models. However, the existing FCNNs could not cope well with sequential learning tasks. In this study, we introduce incremental FCNNs (IFCNNs), which could dynamically update themselves whenever new learning data (e.g., single datum or block data) are incorporated into the dataset. Specifically, we employ dynamic (incremental) fuzzy C-means (FCMs) clustering algorithms to reveal a structure in data and divide the entire input space into several subregions. In the aforementioned partition, the dynamic FCM adaptively adjusts the position of its prototypes by using sequential data. Due to the time-sharing arrival of training data, compared with batch learning models, incremental learning methods may lose classification (prediction) accuracy. In order to tackle this challenge, we utilize quasi-fuzzy local models (QFLMs) based on modified Schmidt neural networks to replace the popular linear functions in TS-type fuzzy models to refine and enhance the ability to represent the behavior of fuzzy subspaces. Meanwhile, the recursive least square error (LSE) estimation is utilized to update the weights of QFLMs from one-by-one or block-by-block (fixed or varying block size) learning data. In addition, the$L_{2}$regularization is considered to ameliorate the deterioration of generalization abilities caused by potential overfitting when carrying out weight estimation. The proposed method leads to the construction of FCNNs in a new way, which can effectively deal with incremental data as well as deliver sound generalization capability. Extensive machine-learning datasets and a real-world application are employed to show the validity and performance of the presented methods. From the experimental results, we show that the proposal can maintain sound classification accuracy when effectively processing sequential data.

Intraclass heterogeneity of medical diagnostic objects poses a challenge for accurate intraclass classification of medical fine-grained images (MFGIs) within deep learning. To accurately classify MFGIs, we propose a novel approach termed fuzzy channel filter-based separable convolution neural networks (FSCNN). The original design of FSCNN comprises the following components: 1) Designing the fuzzy channel filter (FCF) module, devised to establish long-distance feature dependencies for each feature channel with the input image by formulating fuzzy rules “IF–THEN”. 2) The FCF-based separable convolution (FSC) block uses depth-wise and point-wise convolutions to extract and mix feature channels. Then, the internal information of each feature channel is reintegrated through fuzzy weighted averaging in FCF to enhance fine-grained feature information. 3) Creating the deep fuzzy learning architecture FSCNN through the superimposition of FSC blocks. This architectural arrangement enables more effective learning of fine-grained feature distinctions within MFGIs, thereby enhancing classification accuracy. Compared to other advanced fine-grained classification models, including state-of-the-art models, our model outperforms by 2%–6% and 3%–9% on brain MRI and pneumonia CT datasets, respectively.

In this article, we present a dimensionality reduction methodology of reinforced fuzzy-rule-based neural networks (FRNNs) realized with the help of determination/correlation coefficient-based streamlined feature selection strategy and fuzzy clustering with standard deviation to cope with high-dimensional data. This approach aims to reduce the design process of the proposed networks and to curb the computational overhead inherently associated with the increasing volume of data both in terms of their number and the dimensionality of the feature space. The overall architecture and learning mechanism of the FRNNs are based on radial basis function neural networks. However, we design the hidden layer of RBFNNs differently by using fuzzy clustering, which makes it easy to determine the parameters, such as centers and widths of the receptive fields (activation functions). Unlike conventional neural networks, the RBFNNs do not have a feature to support dimensionality reduction. To overcome this limitation, FRNNs select input variables by evaluating the adjusted determination coefficient of the model. To reduce the computational burden of finding an appropriate combination of inputs, we propose a simplified feature selection and elimination technique, in which the variables are selected or eliminated by correlation coefficients. A linear function expresses the connection weight, and we apply L2-norm regularization to least-square-error-based learning to estimate stable coefficients (weights), which is expected to significantly improve the generalization ability. The superiority of the proposed FRNNs was demonstrated by using 28 real-world benchmark datasets. The networks are also compared with the conventional models associated with the FRNNs and several related models previously published in the literature.

This study presents an innovative approach to the design of a hybrid fuzzy classifier, with a focus on exploring the classification capability of a conventional fuzzy polynomial neural network (CFPNN). The proposed novel self-organizing hybrid fuzzy polynomial NN classifier (HFPNNC) improves performance while maintaining model interpretability and fixability by synergistically combining an adaptive network structure and the compound regularization technique (CRT). Recent studies have focused on exploring the potential of CFPNN structures for addressing regression issues. To effectively introduce the CFPNN framework to multiclassification tasks, a resilient fuzzy polynomial neural network structure was designed as a basic subclassifier to construct the proposed HFPNNC. The proposed HFPNNC employs a dynamically adaptive structure comprising of two types of critical layers: 1) the fuzzy set-based polynomial neurons layers and 2) the polynomial neurons layers. This allows the classifier to adapt to the complexity of classification tasks. To further strengthen the robustness and generalization ability of the HFPNNC, we incorporate the synergistic combination of probabilistic constrained competitive response selection (PCCRS) and ℓ2-norm regularization least squares estimation (ℓ2-LSE) methods to manage the generation of network layers and the estimation of neuron weights. As key constituents of the CRT approach, the PCCRS and ℓ2-LSE method strike a balance between model complexity and performance. The effectiveness of the proposed HFPNNC is thoroughly evaluated against classical classifiers, state-of-the-art fuzzy classifiers and deep learning baseline models using 17 public datasets, two real-world datasets, and three large-scale datasets. HFPNNC achieves the best prediction in 72.7% of the data. The experimental results and the statistical analysis show a remarkable advantage of the HFPNNC over existing methods, confirming its potential as a flexible, interpretable solution for classification tasks.

This article is concerned with designing a tobacco leaves classifier through fuzzy clustering-based neural networks, which leverage multiple histogram analyses of images. The key issue of the study is to recognize high-quality and low-quality tobacco leaves only by using color images obtained from real industrial areas. This study applies multiple histogram analyses from different color spaces as image preprocessing to extract the meaningful features from high-resolution images. Dimensionality reduction is performed through principal component analysis to extract essential features to reduce model complexity and alleviate overfitting problems. In a classifier, we apply fuzzy clustering-based neural networks that incorporate fuzzy clustering techniques, especially fuzzy C-means clustering, along with a cross-entropy loss function and its learning mechanism. The process of setting and training the membership function of node in the hidden layer is substituted with fuzzy C-means clustering. Also, Softmax function produces the model's output in terms of class probabilities. The cost function of the networks is determined using the cross-entropy loss function, while the learning process involves Newton's method-based iterative nonlinear least square error estimation. The experiment validates the competitiveness of the proposed design methodology using real tobacco images obtained from the industry. The performance of the proposed classifier is compared against other classifiers previously reported in the literature to demonstrate its effectiveness.

In this article, we introduce a design methodology of reinforced fuzzy models based both on univariate analysis and multivariable analysis to cope with high-dimensional problems. This approach is aimed at reducing the design process and curbing computing overhead inherently associated with the increasing volume of data in terms of both their number and the dimensionality of the feature space. The critical features of the proposed fuzzy models are highlighted as follows: First, the essential input variables of the model are selected by running the univariable and multivariable analyses. In univariate analysis, input variables with a strong linear relationship with the output variable are selected through correlation analysis completed for each input space and output space. On the contrary, in multivariable analysis, input variables are chosen by comparing the determination coefficients obtained from the subsets of input variables. Second, according to the analysis of the input variable, we construct two different kinds of fuzzy models. The first fuzzy model comprises the design of the univariable-based fuzzy model (UFM) and its aggregation. The UFMs are made by the individual input variables selected from the univariable analysis using a correlation coefficient. The subspaces formed by correlation analysis are applied for determining the centers of the membership function (MF). The results produced by individual fuzzy models are aggregated through somet-conorms. The second fuzzy model is with the fuzzy clustering for improving the form of fuzzy space in the premise part of a fuzzy rule. To curb the dramatic increase in size of fuzzy rule in high-dimensional problems, the clustering space is employed as the fuzzy space, and the partition matrix produced by fuzzy provided the required degrees of the MF. Experimental studies include a suite of synthetic and publicly available data. The superiority of the proposed design methodology was demonstrated by using 34 publicly available datasets and also compared with the conventional models associated with the fuzzy rule-based models as well as the state-of-the-art models reported in the literature.

In this article, a reinforced iterative fuzzy radial basis function neural networks (IFRBFNN) is introduced as an augmented FRBFNN architecture generated through the iterative refinement process of weighted fuzzy C-means (WFCM) clustering and weighted least square error estimation. It is well known that the location of the clusters have an effect on the classification performance of the FRBFNN. The underlying idea behind this article is how to define the center points of clusters based on the data distribution analysis as well as the improvement of classification performance. In a nutshell, while the fuzzy C-means clustering to define the positions of radial basis functions (RBFs) based in the unsupervised learning manner is usually used in a FRBFNN, the parameters (cluster centers and their ensuing polynomial coefficients) refinement of FRBFNN through the proposed iterative method lead to superb classification performance by relocating the positions of RBFs over the input space implied by supervised learning. The idea of the proposed approach is to relocate the centers (prototypes) of the fuzzy clusters by using WFCM clustering algorithm and to re-estimate the coefficients by weighted least square estimation with the aid of the cross-entropy loss values of data so that the classification performance is improved. The weight related to each data is determined by some auxiliary information (i.e., the modified version of the cross-entropy loss value of each data). When it comes to the estimation of the coefficients of the consequent polynomial in IFRBFNN, they are estimated by the weighted least square error estimation technique, where the weights of the coefficients are defined based on the cross-entropy loss values. The auxiliary information such as the weights for relocation of the centers of RBFs and the weights for the re-estimation of the polynomial coefficients should be defined from the viewpoint of enhancement of the classification performance. The cross-entropy loss value of each data is able to meet the necessity of the required supervision signal. Several numerical experiments are provided to demonstrate the usefulness of the proposed design method for the FRBFNN for classification problems.

The two issues on dynamically generated hierarchical neural networks such as the sort of basic neurons and how to compose a layer are considered in this article. On the first issue, a variant version of the least-square support vector regression (SVR) is chosen as a basic neuron. Support vector machine (SVM) is a representative classifier which usually shows good classification performance. Along with the SVMs, SVR was introduced to deal with the regression problem. Especially, least-square SVR has the advantages of high learning speed due to the substitution of the inequality constraints by the equality constraint in the formulation of the optimization problem. Based on the least-square SVR, the multiple least-square (MLS) SVR, which is a type of a linear combination of least-square SVRs with fuzzy clustering, is proposed to improve the modeling performance. In addition, a hierarchical neural network, where the MLS SVR is utilized as the generic node instead of the conventional polynomial, is developed. The key issues of hierarchical neural networks, which are generated dynamically layer by layer, are discussed on how to retain the diversity of the nodes located at the same layer according to the increase of the layer. In order to maintain the diversity of the nodes, various selection methods such as truncation selection and roulette wheel selection (RWS) to choose the nodes among candidate nodes are proposed. In addition, in order to reduce the computational overhead to determine all candidates which exhibit all compositions of the input variables, a new implementation method is proposed. From the viewpoint of the diversity of the selected nodes and the computational aspects, it is shown that the proposed method is preferred over the conventional design methodology.

Neural networks (NNs) with least square error (LSE) estimation form a certain type of single hidden layer feed-forward NNs. In this class of networks, the input connections (weights) and the biases of hidden neurons are generated randomly and fixed after being generated. The output connections are estimated by the LSE method rather than the back-propagation method. The random generation of the input connection weights and the hidden biases results in the larger number of hidden neurons to assure the quality of classification performance. To reduce the number of neurons in the hidden layer while maintaining the classification performance, we apply a “divide and conquer” strategy in this article. In other words, we divide an overall input space into several subspaces by using information granulation technique (Fuzzy C-Means clustering algorithm) and determine the local decision boundaries among related subspaces. A decision boundary defined in the input space can be considered as being composed of several decision boundaries defined in subspaces that form the entire input space. For the decision boundaries defined in the subspaces, their nonlinearity becomes lower in comparison with the one being encountered when considering the entire input space. Through the weighted LSE estimation instead of using the LSE estimation method, the connections of several NNs can be estimated without interfering with each other. After estimating the weights, the decision boundaries defined in the related subspaces are merged to a single decision boundary by using fuzzy ensemble technique. Several machine learning datasets and one real world application dataset are used to evaluate and validate the proposed fuzzy ensemble classifier. Based on the experimental results, the proposed classifier shows better classification performance when compared with the performance of some selected classifiers.

In this article, a reinforced fuzzy radial basis function neural network (R-FRBFNN) classifier is proposed. It focuses on the development of methodologies of reinforced architecture to improve classification accuracy and enhance the robust capability based on two learning strategies. The two learning strategies are summarized: 1) R-FRBFNN designed via support vector (SV)-based fuzzy C-means (FCM) clustering and softmax-based iterative reweighted least square (IRLS), which concentrate on improving the classification performance of R-FRBFNN; and 2) R-FRBFNN designed via SV-based FCM and softmax-based iterative quadratic programming (IQP), which focus on improving the robust abilities of the R-FRBFNN and reducing the effects of noise and outliers. The essential points of the proposed R-FRBFNN classifier are summarized as follows. a) The proposed R-FRBFNN consists of three phases: condition, conclusion, and inference. b) An SV-based FCM is considered for prioritizing the classification boundary and improving the classification performance of the proposed classifier. c) Three types of polynomials construct the conclusion phase. Two learning techniques are designed to update the coefficients of the polynomials. Softmax-based IRLS is a type of iterative learning technique based on Newton's method. Softmax-based IQP is more robust and avoids the degradation of generalization capabilities caused by outliers and noisy data. d) In the concept of reinforced architecture, SV-based FCM imposes compensation (membership degrees) on learning techniques according to the data characteristics encountered in the inference phase. Experimental results reported for benchmark data and outliers/noisy datasets demonstrate that the proposed classifier shows improved classification performance compared with other previously studied methods.

In this article, we introduce novel reinforced hybrid fuzzy rule-based neural networks (RHFNNs). This article is concerned with the development of the design methodologies of hybrid fuzzy rule-based model for constructing the network structure and enhancing its predictive abilities through the combination of inhomogeneous neurons [i.e., clustering-based polynomial neurons (CPNs) and polynomial neurons (PNs)] and tournament selection. The key points of the proposed RHFNN are enumerated as follows: The first layer of the proposed network consists of CPNs. CPN can effectively reflect the complex nonlinear structure encountered in the data space, and refine (granulate) it with the help of the clustering algorithm. Two types of CPNs including hard C-means (HCM) clustering-based polynomial neuron (HCPN) and fuzzy C-means (FCM) clustering-based polynomial neuron (FCPN) are designed. According to the type of CPN used in the first layer, RHFNN can be categorized into two types, namely, RHFNN based on HCPN (HRHFNN) and RHFNN based on FCPN (FRHFNN). We use PNs to construct the second and consecutive layers. PN can identify and approximate the nonlinear relationship among system's inputs and outputs. A tournament-based performance selection (TPS) algorithm stemming from evolutionary computation is used for selection of neuron. TPS not only ensures that the candidate nodes have sufficient fitting ability but also enhances the individual diversity in the node set and provides the abilities to generate better prediction nodes. In addition,L2-norm regularization is considered to reduce the deviation between coefficients and ameliorate overfitting as well as boost generalization ability. The performance of RHFNN is discussed through a variety of publicly available machine learning datasets. From the experimental results, we conclude that RHFNN achieves the best prediction accuracy on 13 of 15 datasets; the statistical analysis also confirms the superiority of RHFNN.

In this paper, we propose reinforced fuzzy clustering-based ensemble neural networks (FCENNs) classifier. The objective of this paper is focused on the development of the design methodologies of ensemble neural networks classifier for constructing the network structure and enhancing the learning methods of fuzzy clustering-based neural networks through the combination of the probabilistic model and its learning mechanism. The proposed FCENNs classifier takes into consideration a cross-entropy error function to improve learning while L2norm regularization is used to reduce overfitting as well as enhance generalization abilities. The essential points of the proposed reinforced FCENNs classifier can be enumerated as follows: First, in the proposed classifier, the cross-entropy error function is used as a cost function; to do this, a softmax function is applied to represent a categorical distribution located at the nodes of the output layer. Second, the learning mechanism is composed of two parts. First, fuzzy C-means clustering forms the connections (weights) of the hidden layer while the connections of the output layer are adjusted with the aid of the nonlinear least squares method using Newton's method-based learning. Third, L2norm-regularization is considered to avoid the degradation of generalization ability caused by overfitting. The learning mechanism similar to ridge regression is realized by adding L2penalty term to the cross-entropy error function. From the viewpoint of performance improvement achieved through the proposed novel learning method, the design methodology for the ensemble neural networks classifier is discussed and analyzed with the aid of a diversity of two-dimensional synthetic data and machine learning datasets.