Neural networks (NNs), type-1 fuzzy logic systems (T1FLSs), and interval type-2 fuzzy logic systems (IT2FLSs) have been shown to be universal approximators, which means that they can approximate any nonlinear continuous function. Recent research shows that embedding an IT2FLS on an NN can be very effective for a wide number of nonlinear complex systems, especially when handling imperfect or incomplete information. In this paper we show, based on the Stone-Weierstrass theorem, that an interval type-2 fuzzy neural network (IT2FNN) is a universal approximator, which uses a set of rules and interval type-2 membership functions (IT2MFs) for this purpose. Simulation results of nonlinear function identification using the IT2FNN for one and three variables and for the Mackey-Glass chaotic time series prediction are presented to illustrate the concept of universal approximation. 1. Introduction Several authors have contributed to universal approximation results. An overview can be found in [1–8]; further references to prime contributors in function approximations by neural networks are in [4, 9–12] and type-2 fuzzy logic modeling in [13–23]. It has been shown that a three-layer NN can approximate any real continuous function [24]. The same has been shown for a T1FLS [1, 25] using the Stone-Weierstrass theorem [3]. A similar analysis was made by Kosko [2, 9] using the concept of fuzzy regions. In [3, 26] Buckley shows that, with a Sugeno model [27], a T1FLS can be built with the ability to approximate any nonlinear continuous function. Also, combining the neural and fuzzy logic paradigms [28, 29], an effective tool can be created for approximating any nonlinear function [4]. In this sense, an expert can use a type-1 fuzzy neural network (T1FNN) [10–12, 30] or IT2FNN systems and find interpretable solutions [15–17, 31–34]. In general, Takagi-Sugeno-Kang (TSK) T1FLSs are able to approximate by the use of polynomial consequent rules [7, 27]. This paper uses the Levenberg-Marquardt backpropagation learning algorithm for adapting antecedent and consequent parameters for an adaptive IT2FNN, since its efficiency and soundness characteristics make them fit for these optimizing problems. An Adaptive IT2FNN is used as a universal approximator of any nonlinear functions. A set of IT2FNNs is universal if and only if (iff), given any process , there is a system such that the difference between the output from IT2FNN and output from is less than a given . In this paper the main contribution is the proposed IT2FNNs architectures, which are shown to be universal approximators
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