# WRNFS: Width Residual Neuro Fuzzy System, a Fast-Learning Algorithm with High Interpretability
## 1 Introduction  
Neural networking is an important method in the development of artificial intelligence and an effective tool in the research of brain-like intelligence [1]. Deep neural networks (DNN) have a large number of adjustable free parameters, and the model construction has high flexibility [2]. [PERSON] applied a deep convolutional neural network in the ImageNet contest to solve the image classification task, which creatively implemented a GPU in the convolutional operation [3]; however, due to the excessive number of parameters, a large number of parameters do not have practical significance, resulting in poor interpretability of the model and lack of theoretical support for the process of adjusting parameters; thus, [PERSON] called for the innovation of building a fully-interpretable system rather than making the black box model meaningful [4]. A fuzzy system is a logical reasoning system based on natural language abstraction, which has a great advantage in interpretability [5]. Fuzzy system abstracts fuzzy practical problems into fuzzy functional problems. Membership functions is used to generate mappings based on IF-THEN rules in a fuzzy rule set, which can approximate any non-linear function and effectively combine data information with expert beliefs [6]. [PERSON] proposed the 'fuzzy' ideology, which gives math more capability to deal with a complex real system [7]. Combining with If-Then rules, the fuzzy interference system initiates the human language system better than many other learning algorithms. An artificial intelligence system with good interpretability has a few parameters, among which are mainly based on expert beliefs, and most parameters have practical meaning [8]. Interpretability means that human intelligence can reasonably explain the relationship between parameter values and structure, and efficiently gain experience in the form of natural language [9]. The fuzzy inference system is currently considered an interpretable system with good accuracy [10]; however, fuzzy systems cannot avoid the \"dimension disaster\" problem for high-dimensional problems, suffering from low efficiency and poor robustness [11]. The direct reason is the fuzzy inference system is based on a fuzzy rule set, which grows inevitably in an exponential scale when the input number grows [12]. A small improvement in accuracy usually sacrifices a lot of efficiency and interpretability, through which the system becomes big and lousy, and parameters proliferate rapidly. Current research on fuzzy systems tried to fix this problem using different cluster methods to reduce dimension in the input end, such as fuzzy c-means clustering (FCM), subtractive clustering (SC), and grid partitioning clustering (GPC), etc. [13]. FCM, SC and GPC-based ANFIS realized fuzzy interference systems which can handle high-dimensional problems. In the meantime, the cluster methods reduced the interpretability of the whole systems because the data lost information and their direct meanings in the process. ANFIS is a widely used fuzzy system based on the T-S fuzzy model, which is proved able to approximate arbitrary nonlinear systems [14]. To deal with high-dimensional data sets, FCM-ANFIS, SC-ANFIS, and GPC-ANFIS are applied to many practical problems; however, after dimension reduction, the ANFIS system reduces the interpretability and greatly affects the utilization of input data, which are greatly affected. Several linear functions can approximate an arbitrary nonlinear function [15]. During the approximation, the difference between the target function and the approximated function, called residual, will be the training target of the next generation. The residual network uses residual as the medium between small systems, making manipulating information flow easier, limiting the scale of error in each step, and leaving every small system nearly independent [16]. The idea of residual improves the stability of the whole system and reduces the accuracy requirement of a single layer.