Emerging deep fuzzy neuralnetwork-based fault diagnosis (FD)—integrating deep neural networks (DNNs) and fuzzy neural networks (FNNs) in sequential or parallel architectures—has demonstrated great potential in both high performance and model interpretability. However, in the cascade architecture, the membership functions of the FNN struggle to finely describe the complex fault features extracted by the DNN, leading to reduced FD performance. Meanwhile, the abstraction of fault features undermines the FNN's interpretability. In the parallel architecture, designing a reasonable strategy to fuse the heterogeneous features extracted by the DNN and FNN is nontrivial. To address these challenges, we propose a novel FD model called linguistically interpretable fuzzy neural network with knowledge distillation, which integrates DNN and FNN through the perspective of knowledge distillation, aiming for high performance while preserving FNN's intrinsic interpretability. The model extracts modality probability knowledge from a well-trained DNN teacher unit and feeds into the knowledge distillation unit to generate soft labels encapsulating modality similarity knowledge. These soft labels carry the teacher unit's core insights into FD and activate implicit information of negative modalities. Then, the knowledge-based Takagi–Sugeno–Kang unit uses the soft labels as consequent variables to perform linguistically interpretable rule reasoning from original features to fault modalities. The model is optimized by minimizing a composite loss comprising cross-entropy, soft label regularization, and L2 regularization, ensuring more knowledge is transferred to the distilled unit. Comprehensive evaluation across a series of industrial process cases validated the model's effectiveness in performance and interpretability.

Mainstream strategies for addressing imbalanced monitoring data distributions, such as data sampling and cost-sensitive learning methods, face multiple challenges, including data heterogeneity, uncertainty, biased attention toward normal or fault classes, and reduced discriminability of fault features. These issues collectively lead to a degradation in fault diagnosis (FD) performance. To overcome the aforementioned bottlenecks, this article proposes a unified FD framework, termed a granular computing.based deep neural network (GDNN), which is designed from the novel perspective of granular ball (GB) embedding. Within this framework, the input data are first transformed into suboptimal latent features through a generic deep neural network (DNN).based feature extraction component. Subsequently, an information granulation component reorganizes these features into class-specific local neighborhood structures in the form of GBs. Finally, a Softmax-based fully connected classifier performs FD. The GDNN framework supports end-to-end learning by minimizing a composite loss function consisting of the cross-entropy loss and the justifiable granularity (JG) loss. The main contribution of this work lies in a novel GB embedding mechanism that guides DNNs to focus on GB-based structures rather than isolated individual samples in imbalanced scenarios. Each class is represented by a GB that encapsulates its structural knowledge in the latent feature space, thereby ensuring consistent attention across classes and effectively mitigating severe overfitting. Experimental results on two industrial processes demonstrate that the proposed framework performs robustly across diverse imbalanced scenarios and adapts well to different DNN architectures, providing a novel perspective and a generalizable paradigm for imbalanced FD.

Reconstructing complex physical fields from sparse observations remains a central challenge in science and engineering. Conventional data-driven approaches often depend on large, high-quality datasets and typically overlook underlying physical principles, whereas physics-informed neural networks (PINNs) are constrained by the requirement of predefined governing equations, which are frequently difficult to specify or may fail to capture the true system dynamics. This work presents a physics-guided spectral neural network framework that autonomously infers governing partial differential equations (PDEs) directly from data, enabling mathematical formulations to emerge without prior specification. The framework integrates a multiresolution spectral neural architecture to effectively represent intricate patterns and sharp gradients, together with an adaptive loss training strategy that enforces consistency with the inferred dynamics. The proposed approach demonstrates high reconstruction accuracy across diverse benchmark cases, including combustion temperature fields, unsteady wake flows, 3-D turbulence, and geophysical datasets. Overall, the framework provides a systematic methodology for discovering and applying PDEs, offering a generalizable paradigm for reconstructing physical fields from sparse data.

Federated Fuzzy C-Means (FCM) clustering methods have demonstrated success in standard federated learning (FL) environments, but model heterogeneity presents a significant challenge. This heterogeneity hampers communication content selection and makes traditional aggregation methods, such as federated averaging (FedAvg), insufficient for diverse client models. Additionally, handling non-IID data distributions and learning robust feature representations remain unresolved in some existing approaches. To tackle these issues, we propose FFCMD, a federated FCM clustering framework designed to address model heterogeneity while enhancing feature learning and accommodating non-IID data. FFCMD integrates an autoencoder (AE) with a fuzzy C-Means network (FCMN), augmented by mutual knowledge distillation (mKD), to improve data feature learning, communication, and aggregation under heterogeneous conditions. The AE incorporates a clustering-oriented penalty term to learn discriminative features, boosting performance on complex datasets. Rather than direct parameter aggregation, mKD updates the global model using local devices' soft assignments, naturally handling structural variations across client models. Moreover, FFCMD employs discrepancy-aware weighting mechanisms to dynamically align local and global category distributions during soft assignment embedding, effectively addressing the challenges posed by non-IID data. Extensive experiments on multiple public datasets show that FFCMD outperforms state-of-the-art methods, particularly in non-IID scenarios, and excels even in the presence of model heterogeneity.

Federated Fuzzy C-Means (FCM) has received considerable attention due to the increasing need for privacy-conscious data analysis across diverse domains and sources in many real-world applications. Recent developments in federated FCM, however, are still in their infancy and largely unexplored. These methods struggle to handle nonindependent and identically distributed (non-iid) data. Moreover, critical hyperparameters, such as the number of iterations for local updates, are typically set manually, which can significantly affect the performance of federated clustering. To address these challenges, we introduce an Adaptive Federated FCM with an auxiliary model, named AF-FCM. In this approach, prior information from the auxiliary model, along with a proximal term in the local objective, mitigates the effects of the non-iid environment, enhancing both model robustness and effectiveness. Critical hyperparameters are adaptively adjusted using a proposed adaptive particle swarm optimization (APSO) algorithm, guided by a carefully designed fitness function. Within APSO, a nonlinear regression function adjusts the inertia weight, reducing the risk of convergence to local optima. In AF-FCM, global prototypes are refined using momentum gradient descent (MGD). Numerical experiments highlight the effectiveness and generalization performance of AF-FCM across various conditions, including heterogeneity variations, the number of clients, and the number of clusters. Comparative analysis against state-of-the-art federated clustering baseline methods validates the competitive performance of AF-FCM.

Classification tasks involving tabular data often require a balance between exceptional performance and heightened interpretability. To address this challenge, we propose a linguistically interpretable deep fuzzy classification system called FFT-FFR-RBFC. The system employs a Fuzzy Feature Transformation (FFT) unit, formed by employing a stacked architecture of multiple Takagi-Sugeno-Kang (TSK) fuzzy models with non-linear conclusions, to distill high-level fuzzy features from the input data, a Rule-Based Fuzzy Classifier (RBFC) unit to perform classification using these features, while a Fuzzy Feature Reconstruction (FFR) unit in tandem with the FFT to enhance the system's linguistic interpretability by remapping the high-level features back to their original space. The proposed approach is optimized by minimizing a composite loss function that balances classification and reconstruction losses, ensuring a harmonious interplay between performance and interpretability. Comprehensive evaluation across 20 diverse datasets demonstrates that the system's is exceptionally promising, particularly for high-dimensional or large-scale tabular data classification tasks, achieving superior classification performance while maintaining a high degree of interpretability.

Convolutional neural networks (CNNs) have been developed quickly in many real-world fields. However, CNN’s performance depends heavily on its hyperparameters, while finding suitable hyperparameters for CNNs working in application fields is challenging for three reasons: (1) the problem of mixed-variable encoding for different types of hyperparameters in CNNs, (2) expensive computational costs in evaluating candidate hyperparameter configuration, and (3) the problem of ensuring convergence rates and model performance during hyperparameter search. To overcome these problems and challenges, a hybrid-model optimization algorithm is proposed in this paper to search suitable hyperparameter configurations automatically based on the Gaussian process and particle swarm optimization (GPPSO) algorithm. First, a new encoding method is designed to efficiently deal with the CNN hyperparameter mixed-variable problem. Second, a hybrid-surrogate-assisted model is proposed to reduce the high cost of evaluating candidate hyperparameter configurations. Third, a novel activation function is suggested to improve the model performance and ensure the convergence rate. Intensive experiments are performed on image-classification benchmark datasets to demonstrate the superior performance of GPPSO over state-of-the-art methods. Moreover, a case study on metal fracture diagnosis is carried out to evaluate the GPPSO algorithm performance in practical applications. Experimental results demonstrate the effectiveness and efficiency of GPPSO, achieving accuracy of 95.26% and 76.36% only through 0.04 and 1.70 GPU days on the CIFAR-10 and CIFAR-100 datasets, respectively.

Knowledge graphs are regarded as structured knowledge bases that embody various facts coming from the real world. Their completeness is still far from satisfactory. Relational learning models in link prediction can automatically find the missing relationships between entities to increase the integrality of the knowledge bases, which form two categories purely embedding-based and hybrid embedding-based. Several models including HolE and RotatE belong to purely embedding-based with inefficient performers and few element interactions. Based on the above, this paper advances a novel Knowledge Graph Embedding relational model that leverages a circular correlation operation in the complex domain and dubs as CircularE, which increases interactions between entities and relations to a great extent by this compressed operator without expanding the dimension of space. It gives expression of the interactions between element semantics to achieve good performance in relational learning. Besides, a self-adaption adversarial negative sampling scheme is proposed on account of the KGs structure and the probability semantic of the triples. This negative sampler efficiently optimizes the knowledge representation capability of CircularE and far more than enhances the outputs of several relational original models in embedding-based. Experiments indicate that the competitive properties of CircularE on the four large-scale benchmarks of knowledge base completion tasks are superior to the state-of-the-art methods.

Recently, granular models have been highlighted in system modeling and applied to many fields since their outcomes are information granules supporting human-centric comprehension and reasoning. In this study, a design method of granular model driven by hyper-box iteration granulation is proposed. The method is composed mainly of partition of input space, formation of input hyper-box information granules with confidence levels, and granulation of output data corresponding to input hyper-box information granules. Among them, the formation of input hyper-box information granules is realized through performing the hyper-box iteration granulation algorithm governed by information granularity on input space, and the granulation of out data corresponding to input hyper-box information granules is completed by the improved principle of justifiable granularity to produce triangular fuzzy information granules. Compared with the existing granular models, the resulting one can yield the more accurate numeric and preferable granular outcomes simultaneously. Experiments completed on the synthetic and publicly available datasets demonstrate the superiority of the granular model designed by the proposed method at granular and numeric levels. Also, the impact of parameters involved in the proposed design method on the performance of ensuing granular model is explored.

Data classification problem in the field of machine learning and pattern recognition is an important research content, but the classic data classification algorithms mainly use numeric as the basis of classification modeling or put the high accuracy as the only index of classification model, and these classification algorithms usually are unable to construct classification model for easy understanding. As a basic abstract structure supporting human-centered granular computing methods, information granules can construct easy-to-understand geometric structures for different types of data. In order to solve the above problems, this paper proposes a classification algorithm based on AFS(axiomatic fuzzy set) and information granules to enhance the interpretability of models so that people can better understand the relationship between different types of data. The model can be divided into four stages. In the first stage, each class of data is divided into different data subsets. In the second stage, the AFS membership degrees of the corresponding prototypes are calculated based on axiomatic fuzzy set theory and a series of data blocks are generated on each class based on these degrees. The third stage mainly generates supersphere information granules according to the reasonable granularity principle theory and confidence levels. In the final stage, the activation level is calculated by the distance between samples and the hypersphere information granules and confidence levels to determine the "If-Then" rules.

In time-series forecasting, it is an important task to make an accurate and interpretable long-term prediction. In this article, we present a novel approach developed from the perspective of granular computing (GrC) to realize the long-term prediction of time series. The proposed method first employs a sliding window strategy to smooth on the raw time series. Subsequently, the smoothed time series is transformed into the corresponding granular time series that is depicted by evolving shape with the aid of the clustering algorithm based on the dynamic time warping (DTW) distance. Finally, a Takagi–Sugeno (TS) architecture-like granular model (GrM) is formed by deriving the relations implying in the granular time series and offers the granular output in the numeric vector format. The GrM adopts the pattern-to-pattern inference mechanism to realize the long-term prediction of time series at the vector level. Experiments on several datasets demonstrate that the proposed method not only has the ability to circumvent the cumulative error but also makes the resulting GrM equip better interpretability.

The change of amplitude and frequency result in a variety of variation modality of time series in the universe. It is difficult to describe the variation features of time series exactly relying solely on a single simulating model. To overcome this limitation, a new prediction model using fuzzy cognitive maps is proposed based on partitioning strategies. Initially, fuzzy c-mean clustering is adopted to partition time series into several sub-sequences. Consequently, each partition has its corresponding sequences. Subsequently these sub-sequences are used to constructed fuzzy cognitive maps models respectively. Finally, the fuzzy cognitive maps models are merged by fuzzy rules. The constructed model is not only performing well in numerical prediction but also has interpretability. The experimental results show that the model based on partition strategy is superior to the single.

Numerous learning methods for fuzzy cognitive maps (FCMs), such as the Hebbian-based and the population-based learning methods, have been developed for modeling and simulating dynamic systems. However, these methods are faced with several obvious limitations. Most of these models are extremely time consuming when learning the large-scale FCMs with hundreds of nodes. Furthermore, the FCMs learned by those algorithms lack robustness when the experimental data contain noise. In addition, reasonable distribution of the weights is rarely considered in these algorithms, which could result in the reduction of the performance of the resulting FCM. In this article, a straightforward, rapid, and robust learning method is proposed to learn FCMs from noisy data, especially, to learn large-scale FCMs. The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem in which the least-squares term ensures the robustness of the well-learned FCM and the maximum entropy term regularizes the distribution of the weights of the well-learned FCM. A series of experiments covering two frequently used activation functions (the sigmoid and hyperbolic tangent functions) are performed on both synthetic datasets with noise and real-world datasets. The experimental results show that the proposed method is rapid and robust against data containing noise and that the well-learned weights have better distribution. In addition, the FCMs learned by the proposed method also exhibit superior performance in comparison with the existing methods.

As an augmentation of classic fuzzy models, granular fuzzy models (GFMs) have been applied to many fields being in rapport with experimental data, models, and users. However, most of the existing methods used to construct GFMs are based on the principle of optimal allocation of information granularity, which requires that a numeric model be provided in advance. In this paper, a straightforward and convincing modeling method is proposed to directly construct GFM on a basis of experimental data. The method first granulates the output space to form some interval information granules with distinct semantics and then uses them to partition the entire input space into a series of input subspaces. Subsequently, an initial GFM is emerged by using "If-Then" rules to relate with those interval information granules positioned in the output space and structures expressed in prototypes that are produced by clustering individual input subspaces. Further, the initial GFM is also refined by continuously migrating prototypes in individual input subspaces. The experimental studies using the synthetic dataset and several real-world datasets are reported. They offer a useful insight into the feasibility and effectiveness of the proposed modeling method and reveal the impact of parameters on the performance of the ensuing GFMs. An application example is also presented to exhibit the advantages of the resulting GFM.

Information granules are fundamental, abstract, and easy-to-operate constructs supporting the human-centered handling way in granular computing (GrC). One of the basic properties of information granules comes with its hierarchy. Information granularity is the quantification expression of hierarchy of an information granule. Forming information granules with multigranularity to hierarchically describe the nature of data is an important task in GrC. In this article, a cone-shaped fuzzy set-based granular description method with multigranularity is proposed for multidimensional data. Its fundamental idea is to realize the synergy of quantification of information granularity and a hierarchical description of data by means of α-cuts of cone-shaped fuzzy sets. The proposed method first partition the entire data set into a series of data chunks. Then, some mutually nonoverlapping cone-shaped fuzzy sets are constructed by optimally determining their cores and support radii in terms of the coverage of α-cuts of these fuzzy sets to the data located in individual chunks and the corresponding specificity. Finally, by implementing α-cuts processing for these constructed cone-shaped fuzzy sets, a collection of families of hyper-spherical information granules used to hierarchically describe the structural characteristics of the data set are completely emerged. Besides, the quality of the resulting families of hyper-spherical information granules is evaluated along the granular perspective and the application perspective, respectively. A series of experimental studies concerning several synthetic and publicly available data sets are covered. The experimental results demonstrate the superiority of the proposed granular description method of data with multigranularity.

The imputation of missing values is an important research content in incomplete data analysis. Based on the auto associative neural network (AANN), this paper conducts regression modeling for incomplete data and imputes missing values. Since the AANN can estimate missing values in multiple missingness patterns efficiently, we introduce incomplete records into the modeling process and propose an attribute cross fitting model (ACFM) based on AANN. ACFM reconstructs the path of data transmission between output and input neurons and optimizes the model parameters by training errors of existing data, thereby improving its own ability to fit relations between attributes of incomplete data. Besides, for the problem of incomplete model input, this paper proposes a model training scheme, which sets missing values as variables and makes missing value variables update with model parameters iteratively. The method of local learning and global approximation increases the precision of model fitting and the imputation accuracy of missing values. Finally, experiments based on several datasets verify the effectiveness of the proposed method.

The learning of fuzzy cognitive maps (FCMs) is a timely issue pursued by numerous researchers. Many learning methods, such as population-based algorithms and some hybrid algorithms, have been developed and applied to many fields resulting in better performance. However, those learning algorithms also exhibit obvious limitations: some of them either are difficult to handle the learning problem of large-scale FCM or extremely time-consuming with high computational overload. Furthermore, the learning problem of FCM with noisy data is rarely considered in existing algorithms. In this article, a fast and efficient method for learning FCM is proposed. It first transforms the learning problem of the FCM into a convex optimization problem with constraints, and then, the classic interior-point methods are invoked to solve the optimization problem to obtain the optimized weight matrix of the FCM. A series of experiments involving synthetic data, noisy synthetic data, real data, and publicly available data with noise are considered to demonstrate that the proposed method can rapidly and efficiently learn small-scale and large-scale FCMs and deal with the problem of learning FCM from noisy historical data.

Interval-valued time series (ITS) are interval-valued data that are collected in chronological order. The modeling of ITS is an ongoing issue in domain of time series analysis. This paper presents a new modeling method of ITS based on the synergy of fuzzy set theory and artificial neural networks. The proposed method involves the construction of collection of fuzzy sets describing characteristics of amplitude of ITS, the expression and reconstruction mechanism of ITS and the emergence of model of ITS based on artificial neural network (ANN). The resulting model of ITS not only supports the linguistic output but also the numeric output in interval format. A series of experimental studies is reported for two publicly available financial datasets showing different dynamic characteristics. Experimental results clearly show that the constructed ITS model has the better performance on the linguistic and numeric level.

At present, the development of most of the granular fuzzy models depends upon some well-established numeric ones. In this study, a layered approach used to directly construct granular fuzzy models based on multidimensional numeric data is presented by engaging design methodology of granular computing. The crux of the approach involves a construction of interval information granules in the output space and the corresponding hyperbox information granules in the input space. A method of constructing these information granules and the hyperbox-based granular fuzzy model formed around them is studied in detail. Two different schemes to decode the formed hyperbox-based granular fuzzy model are also presented. Furthermore, a measure of a composite quality of the formed hyperbox-based granular fuzzy model is proposed along with the concept of coverage and specificity of resulting information granules. A number of experimental studies are reported, which offer a useful insight into the effectiveness of the presented approach, as well as reveal the impact of critical parameters on the performance of the established models.

Incomplete data clustering is often encountered in practice. Here the treatment of missing attribute value and the optimization procedure of clustering are the important factors impacting the clustering performance. In this study, a missing attribute value becomes an information granule and is represented as a certain interval. To avoid intervals determined by different cluster information, we propose a congeneric nearest-neighbor rule-based architecture of the preclassification result, which can improve the effectiveness of estimation of missing attribute interval. Furthermore, a global fuzzy clustering approach using particle swarm optimization assisted by the Fuzzy C-Means is proposed. A novel encoding scheme where particles are composed of the cluster prototypes and the missing attribute values is considered in the optimization procedure. The proposed approach improves the accuracy of clustering results, moreover, the missing attribute imputation can be implemented at the same time. The experimental results of several UCI data sets show the efficiency of the proposed approach.

In this paper, a fuzzy c-means clustering algorithm based on interval-valued weights is proposed for improving clustering performance. In the proposed algorithm, the interval-valued weights are first constructed by synergy of the ReliefF algorithm and the analytic hierarchy process (AHP) method, and then they are transformed into a constraint condition associating with each weight variable in the weighted clustering objective function. In the sequence, the weighted clustering objective function is solved by combining the Lagrange multiplier method with the gradient-based iteration computation. In the whole process of algorithm iteration, a compulsion strategy with human–computer cooperation is adopted to ensure each weight variable satisfies interval constraint itself. Three well-known data set are used to perform profound experiments. Experimental results clearly show that the proposed algorithm has better clustering performance than other the weighted fuzzy c-means clustering algorithm.

In this study, a two-stage method which extracts fuzzy rules directly from samples is proposed for classification. First, we introduce a neighborhood based attribute significance algorithm to select r of the most important attributes from the original attribute set. Second, the proposed algorithm generates fuzzy rule from each sample described by the selected attribute subset and finally simplifies the returned fuzzy rule-base. A confidence degree is assigned for each of the extracted fuzzy rules by counting the number of training samples covered by the rule to solve the conflicts among the rules and then the rule-base is pruned. The performance of the proposed classification method have been compared with other five classification approaches including C4.5, DTable, OneR, NNge, and PART on seven UCI data sets. The experimental results show that the proposed method is better than other methods in two aspects: the higher classification accuracy and the smaller rule-base.

In spite of the truly remarkable diversity of models of time series, there is still an evident need to develop constructs whose accuracy and interpretability are carefully identified and reconciled subsequently leading to highly interpretable (human-centric) constructs. While a great deal of research has been devoted to the design of nonlinear numeric models of time series (with an evident objective to achieve high accuracy of prediction), an issue of interpretability (transparency) of models of time series becomes an evident and ongoing challenge. The user-friendliness of models of time series comes with an ability of humans to perceive and process abstract constructs rather than dealing with plain numeric entities. In perception of time series, information granules (which are regarded as realizations of interpretable entities) play a pivotal role. This gives rise to a concept of granular models of time series or granular time series, in brief. This study revisits generic concepts of information granules and elaborates on a fundamental way of forming information granules (both sets—intervals as well as fuzzy sets) through applying a principle of justifiable granularity encountered in granular computing. Information granules are discussed with regard to the granulation of time series in a certain predefined representation space (viz. a feature space) and granulation carried out in time. The granular representation and description of time series is then presented. We elaborate on the fundamental hierarchically organized layers of processing supporting the development and interpretation of granular time series, namely (a) formation of granular descriptors used in their visualization, (b) construction of linguistic descriptors used afterwards in the generation of (c) linguistic description of time series. The layer of the linguistic prediction models of time series exploiting the linguistic descriptors is outlined as well. A number of examples are offered throughout the entire paper with intent to illustrate the main functionalities of the essential layers of the granular models of time series.

Most researchers of time series forecasting devote to design and develop quantitative models for pursuing high accuracy of forecasting on the numerical level. However, in real world, the numerical accuracy is sometimes not necessary for human cognition and decision-making and the numerical results of forecasting based on quantitative model are deficient in interpretability, thus the development of qualitative forecasting model of time series becomes an evident challenge. In this paper, the improved fuzzy cognitive map (IFCM) are proposed first, and then it is applied to develop qualitative model for linguistic forecasting of time series together with fuzzy c-means clustering technology and real-coded genetic algorithm (RCGA). Two real life time series are used to test the developed forecasting model and compare with another method based on FCM, whose results show the developed FCM forecasting model is more simpler and high quality on the linguistic level.