# Self-Organized Fuzzy Neural Network Nonlinear System Modeling Method Based on Clustering Algorithm
###### Abstract  
In this paper, an improved self-organizing fuzzy neural network (SOFNN-CA) based on a clustering algorithm is proposed for nonlinear systems modeling in industrial processes. In order to reduce training time and increase training speed, we combine offline learning and online identification. The unsupervised clustering algorithm is used to generate the initial centers of the network in the offline learning phase, and, in the self-organizing phase of the system, the Mahalanobis distance (MD) index and error criterion are adopted to add neurons to learn new features. A new density potential index (DPI) combined with neuron local field potential (LFP) is designed to adjust the neuron width, which further improves the network generalization. The similarity index calculated by the Gaussian error function is used to merge neurons to reduce redundancy. Meanwhile, the convergence of SOFNN-CA in the case of structural self-organization is demonstrated. Simulations and experiments results show that the proposed SOFNN-CA has a more desirable modeling accuracy and convergence speed compared with SOFNN-ALA and SOFNN-AGA. self-organizing fuzzy neural networks (SOFNN); nonlinear system modeling; Mahalanobis distance; density potential index +
Footnote †: journal: Applied Sciences  
rules with the learning ability of neural networks. In practical applications, two important problems to be solved are structure identification and parameter estimation [10]. Structure identification needs to determine the size of the fuzzy rules, and parameter estimation needs to determine the parameters of the fuzzy rules that make the system reliable. The structure identification of FNN is divided into offline or online. Offline recognition updates fuzzy rules and parameters based on the whole training set, and generally uses expert knowledge and clustering algorithms to generate suitable fuzzy rules. There are many algorithms for neural network structure initialization, such as fuzzy c-means clustering, k-nearest neighbor clustering, density-based spatial clustering, and agglomerative clustering. However, current research often develops self-organizing algorithms for online identification and ignores the role of offline identification in the early stages of system modeling. Online recognition means training and updating the weight parameters based on each sample that comes in real time, which satisfies the requirements of online control in industry. Online identification commonly uses self-organizing methods to adjust the number of neurons through a series of structural identification indexes. For example, [PERSON] et al. [11; 12] proposed that the dynamic fuzzy neural network (D-FNN) and its enhanced version of the generalized dynamic fuzzy neural network (GD-FNN) can be used for online structure identification and parameter optimization, but the generalization performance of the system is poor. [PERSON] et al. [13] used the relative importance index of each rule to achieve the self-organization of FNN and [14] used the information theoretic approach to design fuzzy rules by dividing fuzzy rules with high peak intensity into new rules and branching fuzzy rules with small relative mutual information (SOFNN-ACA). However, in the initial stage of online training, it takes a longer time to identify the appropriate result due to the absence of a suitable cluster center, which undoubtedly increases the response time of the system. Therefore, some scholars use deep learning method to initialize the sample and then feed it into a self-organizing fuzzy neural network for learning; for example, [PERSON] et al. [15] designed a self-growing algorithm based on incremental deep pre-training (IDPT) to extract effective features and use them as input to SOFNN, which achieves the extraction of deep features from the original data and effectively speeds up the training.