Bivariate fully fuzzy interpolation problem using artificial neural networks approach

Abstract

Artificial neural networks modeling is one of the most prominent techniques for solving more complicated mathematical problems that can not be solved in the traditional computing environments. The work described here intends to offer an efficient bivariate fuzzy interpolation methodology based on the artificial neural networks approach. It has several notable features including high processing speeds and the ability to learn the solution to a problem from a set of examples which categorizes them in line of intelligent systems. To do this, a multilayer feed-forward neural architecture is depicted for constructing a fully fuzzy interpolating polynomial of arbitrary degree. Then, a back-propagation supervised learning optimization algorithm will be applied for estimating the unknown fuzzy coefficients of the solution polynomial. Finally, the advantage of our technique is illustrated by using some practical examples to show the ability of the improved algorithm in solving rigorous problems.

Keywords

Bivariate fully fuzzy polynomial artificial neural network approximate solution criterion function learning algorithm

1 Introduction

Interpolation is an important concept in the applied mathematical field of numerical analysis for constructing an approximation of solution function at a new set of data points. Interpolation methods have been the subject of an active field of research. Here, the authors have chosen some valuable and useful studies that have been presented to deal with fuzzy interpolative methods. For example, a multi-variable interpolation technique based on fuzzy rule interpolation rules has been presented to the temperature pre- diction problem in [3]. Bai and Zhuang in [2], described an interpolation technique based on a fuzzy error interpolation method, to estimate robot position errors, which was consistent with the random distributed nature of the position errors. Szeghegyi and Bede used some properties of bivariate fuzzy B-spline series and fuzzy splines of Schoenberg-type, to approximate of a bivariate fuzzy function [15]. Li et al. constructed a bivariate interpolation method using artificial neural networks approach that was interpolated as the interpolation data [12]. In [1], having as starting point fuzzy interpolation, the bivariate fuzzy B-spline series are used in digital terrain modeling. Also, a non-linear interpolation technique was offered on sparse fuzzy rule bases with multiple inputs [11].

Neural network simulations, with their striking ability to derive meaning from imprecise data, can be used in solving the problems. In fact, neural nets are self-learning mechanisms that is shows no needs to implement traditional abilities of a programmer. Recently, some neural architectures have been employed to approximate solution of different kinds interpolation problems. In [9], a multi-layer feed-forward network has been successfully applied to model the bivariate interpolation problem on a real set points. Also, a fuzzified architecture of these nets has been designed to estimate the univariate fully fuzzy interpolating polynomial [10]. The aim of this paper is to introduce the fuzzy neural networks approach as an alternative scheme for multi-variable fuzzy rule interpolation. Hence, a trainable four-layer feed-forward architecture of fuzzy neural nets is used to estimate a bivariate fuzzy interpolating polynomial, in which can be thought by a set of fuzzy support mesh points. In the proposed network architecture, the unknown fuzzy interpolation coefficients of solution polynomial are considered as net- work parameters (weights). Now, a back-propagation supervised learning optimization algorithm based on the gradient descent method is achieved on the given fuzzy mesh points by adjusting the connection weights of neurons, to reduce the current error between the reconstructed output and the desired one. This global error reduction is done over training steps by continuously updating the connection weights until an acceptable network accuracy is reached. This work yields a two variables fully fuzzy polynomial to solve the interpolation problem. The material presented below can be divided into four parts: Section 2, describes the proposed fuzzy iterative interpolative reasoning technique; convergence analysis is expressed in Section 3; the effectiveness of the proposed algorithm is illustrated by solving two practical examples in Section 4; finally, conclusion is described in Section 5.

2 Description of the method

Bivariate interpolation which consists of interpolation of a function of two variables, is utilized in many mathematical processes and complicated numerical analysis. It can be viewed that discussing interpolation of a two variables continuous function would pass through all theoretical difficulty in order to set up a framework, where numerical values of this function are known at finite mesh points on two dimensional space. Polynomials with their simple structure, which makes them easy to construct effective approximations, can be addressed as the basic means in the approximation of functions. For this reason, a representation of two variables polynomials approach that is one of the most widely used technique in the context of interpolation, is offered for solving interpolation problem on a multi-variable information space E2 (the set of all fuzzy numbers is denoted by E). Since, the quality of fuzzy numbers have been discussed in many researches, we will pass them by here to prevent reader to be exhausted. More detailed information on this application can be found in [6, 16].

2.1 Problem description

In this part, some fundamental are summarized and powerful interpolation concepts that will be used throughout this paper. The procedure begins with the problem that will be solved in the following. Let {(x_i, y_i) ; z_i} (for i = 0, …, n) be n + 1 fuzzy interpolation nodes, in which the corresponding number z_i represents the value attained at the mesh point (x_i, y_i) by the continuous solution function φ with an appropriate bivariate fuzzy polynomial by using the artificial neural networks approach.

2.2 ANNs approach

Artificial neural networks are intelligent computational techniques in a nonlinear process modeling, that have been taken an important place with the development of utility computing and information processing. These networks are data processing systems that are similarity imitated from the human brain. Here, the data processing is done by lots of parallel processors which are called “neurons”. In these networks, a data structure is designed with the help of programming knowledge, that can act as biological nervous systems. Simply, neural networks are expert systems with new computational methods in which by creating a logical relation between the artificial neurons and then providing a suitable learning algorithm, can solve a special modeled problem. These networks now are applied in many branches of science. Due to their high efficiency in modeling the natural world problems, they have been well studied in many literatures. Because it is not possible to clear all of these studies, and to bring a clear understanding to anyone interested in the subject, the authors have prepared references to refer to [7, 8].

In this part the matter of approximating a given multi variables fuzzy function is addressed by using artificial neural networks. Let us consider (n + 1) (m + 1) nodes {(x_i, y_j) ; z_ij} (for i = 0, …, n ; j = 0, …, m). The main problem is to construct the interpolating polynomial φ_n,m (x, y) for bivariate fuzzy space {(x_i, y_j) ; z_ij} ij of arbitrary degree, such that φ_n,m interpolates z_ij at the node (x_i, y_j), There are many ways to write down mentioned polynomial, but if the interpolation nodes are not uniformly distributed over a Cartesian grid, then the interpolation polynomial is difficult to find. Now, generally speaking, it is preferable to resort to use an alternative rule such as ANNs approach. Denoting the two sets of abscissas defined above by X = x₀, …, x_n and Y = y₀, …, y_m, now the corresponding rectangular mesh points via Cartesian product of the two sets as follows: $X \times Y = {(x_{i}, y_{j})} x_{i} \in X, y_{i} \in Y .$

Now, consider x and y blended polynomial φ_n,m as approximation to solution fuzzy function, that is given in the following double summation form: $φ_{n, m} (x_{i}, y_{i}) = \sum_{i = 0}^{n} \sum_{j = 0}^{m} a_{ij} x^{i} y^{j} .$ (1)

The given power form is the standard way to specify a two variables polynomial in mathematical discussions. Due to the fact that, the derivative and indefinite integral of polynomials are easy to determine, then it will be convenient to o?er the representation formula(1) for the bivariate fuzzy interpolation problem.

Here, in order to get an iterative scheme for estimating bivariate fuzzy interpolation problem (BFIP), a brief framework of the proposed neural networks architecture is offered. Consider the following version of a four-layer feed- forward fuzzy neural architecture shown in Fig. 1. Let us denote the ®-level sets of the fuzzy parameter [o] by: $[o]^{\propto} = [o]_{i}^{\propto} [o]_{u}^{\propto}, \propto \in [0, 1],$

where l and u index the left-hand side and right-hand side of the ∝-level sets of fuzzy number [.], respectively. Also, the input-output relation of this neural architecture is given for the ∝-level sets as follows: $\begin{matrix} * input units : \\ {[o_{1}^{1}]}^{\propto} = [o_{1}^{1}]_{i}^{\propto}, [o_{1}^{1}]_{u}^{\propto} = [x]_{i}^{\propto}, [x]_{u}^{\propto}, \end{matrix}$ (2) ${[o_{2}^{1}]}^{\propto} = [o_{2}^{1}]_{i}^{\propto}, [o_{2}^{1}]_{u}^{\propto} = [y]_{i}^{\propto}, [y]_{u}^{\propto},$

$\begin{matrix} * first hidden units : \\ {[o_{i, 1}^{2}]}^{\propto} & = & [[o_{i, 1}^{2}]_{i}^{\propto}, [o_{i, 1}^{2}]_{u}^{\propto}] = [[x^{i}]_{i}^{\propto}, [x^{i}]_{u}^{\propto}] \\ = & {[min {([x_{i}^{\propto}])}^{i}, ([x_{u}^{\propto}])^{i}}, max {([x_{i}^{\propto}])^{i}, \\ ([x_{u}^{\propto}])^{i}}, i = 1, \dots, n] \\ {[o_{j, 2}^{2}]}^{\propto} & = & [[o_{j, 2}^{2}]_{i}^{\propto}, [o_{j, 2}^{2}]_{u}^{\propto}] = [[y^{j}]_{i}^{\propto}, [y^{j}]_{u}^{\propto}] \\ = & {[min {([y_{i}^{\propto}])}^{j}, [(y_{u}^{α})]^{j}}, max {([y_{i}^{α}])^{j}, \\ ([y_{u}^{\propto}])^{j}}, j = 1, \dots, m] \end{matrix}$ (3)

$\begin{matrix} * second hidden units : \\ [o_{i, 0}^{3}]^{\propto} = [[o_{i, 0}^{3}]_{i}^{\propto}, [o_{i, 0}^{3}]_{u}^{\propto}] = [[a_{i, 0} \cdot o_{i, 0}^{2}]_{i}^{\propto}, \\ [a_{i, 0} \cdot o_{i, 1}^{2}]_{u}^{\propto}] \\ [o_{0, j}^{3}]^{\propto} = [[o_{0, j}^{3}]_{i}^{\propto}, [o_{0, j}^{3}]_{u}^{\propto}] = [[a_{0, j} \cdot o_{j, 2}^{2}]_{i}^{\propto}, \\ [a_{0, j} \cdot o_{j, 2}^{2}]_{u}^{\propto}] \\ [o_{i, j}^{3}]^{\propto} = [[o_{i, j}^{3}]_{i}^{\propto}, [o_{i, j}^{3}]_{u}^{\propto}] = [[a_{i, 1} \cdot o_{j, 2}^{2}]_{i}^{\propto}, \\ [a_{i, 1} \cdot o_{j, 2}^{2}]_{u}^{\propto}] \end{matrix}$ (4)

Where $\begin{matrix} [o_{i, 0}^{3}]_{l}^{\propto} & = & min {[a_{i, 0}]_{l}^{\propto} [o_{i, 1}^{2}]_{l}^{\propto}, [a_{i, 0}]_{l}^{\propto} [o_{i, 1}^{2}]_{u}^{\propto}, \\ [a_{i, 0}]_{u}^{\propto} [o_{i, 1}^{2}]_{l}^{\propto}, [a_{i, 0}]_{u}^{\propto} [o_{i, 1}^{2}]_{u}^{\propto}} \\ {[o_{i, 0}^{3}]}_{u}^{\propto} & = & max {[a_{i, 0}]_{l}^{\propto} [o_{i, 1}^{2}]_{l}^{\propto}, [a_{i, 0}]_{l}^{\propto} [o_{i, 1}^{2}]_{u}^{\propto}, \\ [a_{i, 0}]_{u}^{\propto} [o_{i, 1}^{2}]_{l}^{\propto}, [a_{i, 0}]_{u}^{\propto} [o_{i, 1}^{2}]_{u}^{\propto}} \\ {[o_{i, 0}^{3}]}_{u}^{\propto} & = & min {[a_{0, j}]_{l}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{0, j}]_{l}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}, \\ [a_{0, j}]_{u}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{0, j}]_{u}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}} \\ {[o_{i, 0}^{3}]}_{u}^{\propto} & = & max {[a_{0, j}]_{l}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{0, j}]_{l}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}, \\ [a_{0, j}]_{u}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{0, j}]_{u}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}} \\ {[o_{i, j}^{3}]}_{u}^{\propto} & = & min {[a_{i, 1}]_{l}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{i, 1}]_{l}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}, \\ [a_{i, 1}]_{u}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{i, 1}]_{u}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}} \\ {[o_{i, j}^{3}]}_{u}^{\propto} & = & max {[a_{i, 1}]_{u}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{i, 1}]_{l}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}, \\ [a_{i, 1}]_{u}^{\propto} [o_{j, 2}^{2}]_{l}^{\propto}, [a_{i, 1}]_{u}^{\propto} [o_{j, 2}^{2}]_{u}^{\propto}} \end{matrix}$

Where $\begin{matrix} [{net}_{1}]_{l}^{α} & = & \sum_{i = 1}^{n} [o_{i, 0}^{3}]_{l}^{\propto}, \\ {[{net}_{2}]}_{l}^{α} & = & \sum_{j = 1}^{n} [o_{0, j}^{3}]_{l}^{\propto}, \\ {[{net}_{3}]}_{l}^{α} & = & \sum_{i = 1}^{n} \sum_{j = 1}^{m} [o_{i, j}^{3} \cdot a_{i, j}]_{l}^{\propto}, \\ {[{net}_{1}]}_{u}^{α} & = & \sum_{i = 1}^{n} [o_{i, 0}^{3}]_{u}^{\propto}, \\ {[{net}_{2}]}_{u}^{α} & = & \sum_{j = 1}^{n} [o_{0, j}^{3}]_{u}^{\propto}, \\ {[{net}_{3}]}_{u}^{α} & = & \sum_{i = 1}^{n} \sum_{j = 1}^{m} [o_{i, j}^{3} \cdot a_{i, j}]_{u}^{\propto}, \end{matrix}$

And $\begin{matrix} [o_{i, j}^{3} \cdot a_{i, j}]_{l}^{\propto} = min \\ {[o_{i, j}^{3}]_{l}^{\propto} [a_{i, j}]_{l}^{\propto}, [o_{i, j}^{3}]_{l}^{\propto} [a_{i, j}]_{u}^{\propto}, [o_{i, j}^{3}]_{u}^{\propto} [a_{i, j}]_{l}^{\propto}, \\ [o_{i, j}^{3}]_{u}^{\propto} [a_{i, j}]_{u}^{α}}, \\ {[o_{i, j}^{3} \cdot a_{i, j}]}_{u}^{\propto} = max \\ {{[o_{i, j}^{3}]}_{l}^{\propto} [a_{i, j}]_{l}^{\propto}, [o_{i, j}^{3}]_{l}^{\propto} [a_{i, j}]_{u}^{\propto}, [o_{i, j}^{3}]_{u}^{\propto} [a_{i, j}]_{l}^{\propto}, \\ [o_{i, j}^{3}]_{u}^{\propto} [a_{i, j}]_{u}^{α}}, \end{matrix}$

It should be noted that the extension principle of Zadeh plays particularly important role in enabling us to calculate the relations between input neurons and output one. For further details, refer to [17].

In this part, an architecture of ANNs has been structured for a particular application in which the network must be trained before it becomes useful. To start this procedure, optionally a training or learning begins, which specifies how to adjust the network parameters for a given training pattern. In other words, the network needs to be trained with a learning rule which will be described below. In other words, the network needs to be trained with a learning rule which will be described along with evaluating a given polynomial in above form.

2.2.1 Criterion function

Error correction rules were proposed for network training which essentially drive the output error of a given network to zero. Throughout this part, an attempt is made to define an appropriate criterion function to calculate difference between the current output and the desired output. Consider again the proposed particular network architecture shown in Fig. 1. A commonly error function as the mean-squared error is applied to transform the problem to being minimized on the fuzzy input-output space as follows:

$[e_{p, q}]^{\propto} = [e_{p, q}]_{l}^{\propto} + [e_{p, q}]_{u}^{\propto},$ (6)

Where $\begin{matrix} [e_{p, q}]_{l}^{\propto} = \frac{1}{2} {({[z_{p, q}]}_{l}^{\propto} - {[φ (x_{p}, y_{q})]}_{l}^{\propto})}^{2}, \\ [e_{p, q}]_{u}^{\propto} = \frac{1}{2} {({[z_{p, q}]}_{u}^{\propto} - {[φ (x_{p}, y_{q})]}_{u}^{\propto})}^{2} . \end{matrix}$

Then the total error of the given multi-layer net over all the training patterns, is obtained as: $e = \sum_{\propto} \sum_{p = 0}^{n} \sum_{q = 0}^{m} [e_{p, q}]^{\propto} .$ (7)

2.2.2 Proposed learning rule

Studying this concept that how ANNs work, is fifty of years old. By the end of this part it will be established that our proposed learning rule can be systematically derived as minimizers of the mentioned criterion function. In other words, our interest in the remainder of this paper is in training the proposed network architecture to perform this task. Here, we intend to illustrate the above idea by deriving an appropriate supervised gradient descent-based learning procedure which is a natural generalization of the delta rule, such that can adaptively adjust all weight parameters in the network to reduces the error over a given training set {(x_p, y_q) ; z_p,q}. This work yields the network output converges for each given input to the corresponding desired output. To do this, the new value for each weight parameter is found by taking the current weight and adding an amount that is proportional to the slope of training. In this issue, in order to simplify the computations procedure, the network parameters (inputs, weights and output) selected as fuzzy numbers of trapezoidal form. Assume that, the network is being trained to learn data pairs {(x_p, y_q) ; z_p,q} (for p = 0, …, n ; q = 0, …, m).

Now, the complete procedure for updating the i, j-th connection weight $a_{i, j} = (a_{i, j}^{1}, a_{i, j}^{2}, a_{i, j}^{3}, a_{i, j}^{4})$ , and utilizing mentioned back-propagation learning procedure yields:

$\begin{matrix} a_{i, j}^{r} (t + 1) & = & a_{i, j}^{r} (t) + Δ a_{i, j}^{r} (t), \\ i & = & 0, \dots, n; j = 0, \dots, m, \end{matrix}$ (8)

$\begin{matrix} Δ a_{i, j}^{r} (t) & = & - η \cdot \frac{\partial [e_{p, q}]^{\propto}}{\partial a_{i, j}^{r}} + γ . Δ a_{i, j}^{r} (t - 1), \\ r & = & 1, \dots, 4, \end{matrix}$ (9)

where t is the number of adjustments, η and γ are the small constant learning rate and the momentum term constant in which normally chosen between 0 and 1, respectively. In addition, $a_{i, j}^{r} (t + 1)$ and $a_{i, j}^{r} (t)$ represent the updated and current indexes of fuzzy weight a_i,j, respectively. Since the fuzzy targets for training sets are explicitly specified, one can directly use gradient descent on the criterion function for updating connection weights. Therefore, for gradient descent-based adaptation, the partial derivatives of [e_p,q] ^∝ with respect to the weights a_i,j can now computed by employing the entropy criterion as follows $\frac{\partial [e_{p, q}]^{\propto}}{\partial a_{i, j}^{r}} = \frac{\partial [e_{p, q}]_{l}^{\propto}}{\partial a_{i, j}^{r}} + \frac{\partial [e_{p, q}]_{u}^{\propto}}{\partial a_{i, j}^{r}},$ (10)

Where $\begin{matrix} \frac{\partial [e_{p, q}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & \frac{\partial [e_{p, q}]_{l}^{\propto}}{\partial [φ (x_{p}, y_{q})]_{l}^{\propto}} \cdot \frac{\partial [φ (x_{p}, y_{q})]_{l}^{\propto}}{\partial a_{i, j}^{r}} \\ = & ({[φ (x_{p}, y_{q})]}_{l}^{\propto} - {[z_{p, q}]}_{l}^{\propto}) \cdot \\ \times \frac{\partial [φ (x_{p}, y_{q})]_{l}^{\propto}}{\partial a_{i, j}^{r}}, \\ \frac{\partial [e_{p, q}]_{u}^{\propto}}{\partial a_{i, j}^{r}} & = & \frac{\partial [e_{p, q}]_{u}^{\propto}}{\partial [φ (x_{p}, y_{q})]_{u}^{\propto}} \cdot \frac{\partial [φ (x_{p}, y_{q})]_{u}^{\propto}}{\partial a_{i, j}^{r}} \\ = & ({[φ (x_{p}, y_{q})]}_{u}^{\propto} - {[z_{p, q}]}_{l}^{\propto}) \cdot \\ \times \frac{\partial [φ (x_{p}, y_{q})]_{u}^{\propto}}{\partial a_{i, j}^{r}}, \end{matrix}$

With $\begin{matrix} \frac{\partial {[φ (x_{p}, y_{q})]}_{l}^{\propto}}{\partial a_{i, j}^{r}} = \frac{\partial {[{net}_{1}]}_{l}^{\propto}}{\partial a_{i, j}^{r}} + \frac{\partial {[{net}_{2}]}_{l}^{\propto}}{\partial a_{i, j}^{r}} \\ + \frac{\partial {[{net}_{3}]}_{l}^{\propto}}{\partial a_{i, j}^{r}} + \frac{\partial {[a_{0, 0}]}_{l}^{\propto}}{\partial a_{i, j}^{r}}, \\ \frac{\partial {[φ (x_{p}, y_{q})]}_{u}^{\propto}}{\partial a_{i, j}^{r}} = \frac{\partial {[{net}_{1}]}_{u}^{\propto}}{\partial a_{i, j}^{r}} + \frac{\partial {[{net}_{2}]}_{u}^{\propto}}{\partial a_{i, j}^{r}} \\ + \frac{\partial {[{net}_{3}]}_{u}^{\propto}}{\partial a_{i, j}^{r}} + \frac{\partial {[a_{0, 0}]}_{u}^{\propto}}{\partial a_{i, j}^{r}}, \end{matrix}$

To complete the calculation of derivative terms in above equations, the chain rule for differentiation at the current weight values is employed. So, $\begin{matrix} \frac{\partial [{net}_{1}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & {\begin{matrix} \frac{\partial [o_{i, 0}^{3}]_{l}^{\propto}}{\partial [a_{i, 0}]_{l}^{\propto}}, & j = 0, \\ 0, & j \neq 0 \end{matrix} \\ \frac{\partial [{net}_{2}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & {\begin{matrix} \frac{\partial [o_{0, j}^{3}]_{l}^{\propto}}{\partial [a_{0, j}]_{l}^{\propto}}, & i = 0, \\ 0, & i \neq 0 \end{matrix} \end{matrix}$ $\begin{matrix} \frac{\partial [{net}_{3}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & {\begin{matrix} \frac{\partial [{net}_{2}]_{l}^{α}}{\partial [a_{i, j}]_{l}^{\propto}} \cdot & \frac{\partial [a_{i, j}]_{l}^{\propto}}{\partial a_{i, j}^{r}}, \\ \frac{\partial [{net}_{2}]_{l}^{α}}{\partial [a_{i, j}]_{u}^{\propto}} \cdot & \frac{\partial [a_{i, j}]_{u}^{\propto}}{\partial a_{i, j}^{r}}, \end{matrix} \end{matrix}$ $\begin{matrix} = & {\begin{matrix} \partial [a_{i, j}]_{l}^{\propto}, [o_{i, 0}^{3}]_{l}^{\propto} \geq 0 or [a_{i, j}]_{l}^{\propto} \\ < 0 [o_{i, 0}^{3}]_{u}^{\propto} \geq 0 \\ \partial [a_{i, j}]_{u}^{\propto}, [o_{i, 0}^{3}]_{l}^{\propto} < 0 or [a_{i, j}]_{l}^{\propto} \\ \geq 0 [o_{i, 0}^{3}]_{l}^{\propto} < 0 \end{matrix} \end{matrix}$

Where $\begin{matrix} \frac{\partial {[{net}_{3}]}_{l}^{\propto}}{\partial {[a_{i, j}]}_{l}^{\propto}} & = & {\begin{matrix} [o_{i, 0}^{3}]_{l}^{\propto}, [a_{i, j}]_{l}^{\propto}, [o_{i, 0}^{3}]_{l}^{α} \geq 0 \\ [o_{i, 0}^{3}]_{l}^{\propto}, [a_{i, j}]_{u}^{\propto} \geq 0, [o_{i, 0}^{3}]_{u}^{α} < 0 \end{matrix}, \\ \frac{\partial [a_{i, j}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & {\begin{matrix} 1 - \propto r = 1 \\ \propto r = 2 \\ 0 r = 3 \\ 0 r = 4 \end{matrix} \end{matrix}$ $\begin{matrix} \frac{\partial [{net}_{3}]_{l}^{\propto}}{\partial [a_{i, j}]_{u}^{\propto}} & = & {\begin{matrix} [o_{i, 0}^{3}]_{u}^{\propto}, & [a_{i, j}]_{u}^{\propto}, [o_{i, 0}^{3}]_{u}^{\propto} \geq 0 \\ [o_{i, 0}^{3}]_{u}^{\propto}, & [a_{i, j}]_{l}^{\propto} \geq 0, [o_{i, 0}^{3}]_{u}^{\propto} < 0 \end{matrix}, \\ \frac{\partial [a_{i, j}]_{l}^{\propto}}{\partial a_{i, j}^{r}} & = & {\begin{matrix} 0 r = 1 \\ 0 r = 2 \\ \propto r = 3 \\ 1 - \propto r = 4 \end{matrix} \end{matrix}$

Using a similar procedure as an outline the correspondingly corollary simplifies $\frac{\partial [{net}_{1}]_{u}^{\propto}}{\partial a_{i, j}^{r}}$ , in which we are refrained to go through proof details. It is clearly seen that if n grows, we would like the output φ (x_p, y_q) to be better and better estimate of the desired value z (x_p, y_q).

3 Convergence analysis

In this section, an attempt is made to give a proof for convergence analysis of the proposed iterative method, by presenting the universal approximation property of artificial neural networks. It has been shown in [4] that regular networks can approximate any continuous function on a compact set to any desired degree of accuracy. In other words, the solution function can be estimated with the meaning of uniform topology on E² by input-output relations of designed four-layers fuzzy feed-forward neural architecture.

Theorem 1.Assumef (x, t) : E² → Ebe a continuous bivariate fuzzy function. Then for arbitrary ∈>0, there is an integerRand fuzzy numbersa_i,j (fori = 0, …, n ; j = 0, …, m) such that for allr ≥ R : $\begin{matrix} D (f, φ) & = & sup_{(x, t) \in E^{2}} {max [| \underline{f} (x, y, \propto) \\ - \underline{φ} (x, y, \propto) |, | \bar{f} (x, y, \propto) \\ - \bar{φ} (x, y, \propto) | |]} \leq ɛ, \end{matrix}$

where (E², D) is a complete metric space [14].

Proof. See [5].

The training of the proposed neural architecture is done in such a way that for each input set, the connection weights can be updated in the manner that the calculated output converges for each given input to successive accuracy. It can easily be concluded that, if a solution exists, it is achieved in a finite number of iterations.

4 Experimental results

In order to assess the accuracy and capability of the method, the following two test problems are carried out in this section. From a practical point of view, a relevant use of the proposed iterative algorithm will be applied in two numerical examples. Comparing the obtained results, reveals that our methods is very efficient and useful in the field. All calculations in the following tables are performed using Matlab v7 : 8 Consider the specifications as follows:

1)
Learning rate: η = 0.01,

Momentum constant: γ = 0.01,

Total-error:

$\begin{matrix} [E_{max}]_{l}^{α} & = & (\sum_{i = 1}^{n} \sum_{j = 1}^{m} | {[φ (\frac{x_{i}}{2}, \frac{y_{j}}{2})]}_{l}^{\propto} \\ - {[f (\frac{x_{i}}{2}, \frac{y_{j}}{2})]}_{l}^{\propto} |), \\ {[E_{max}]}_{l}^{α} & = & (\sum_{i = 1}^{n} \sum_{j = 1}^{m} | {[φ (\frac{x_{i}}{2}, \frac{y_{j}}{2})]}_{l}^{\propto} \\ - {[f (\frac{x_{i}}{2}, \frac{y_{j}}{2})]}_{l}^{\propto} |), \end{matrix}$

Where f is the exact fuzzy interpolating function.

Example 4.1. Consider the following functional support nodes: $\begin{matrix} z_{0, 0} & = & (10.0625, 16.0000, 18.5000, 48.00), \\ z_{0, 1} & = & (10.3125, 19.7500, 25.5000, 54.75), \\ z_{0, 2} & = & (9.0625, 25.0000, 33.5000, 70.500), \\ z_{1, 0} & = & (10.0625, 17.5000, 22.5000, 60.00), \\ z_{1, 1} & = & (10.3125, 21.2500, 29.5000, 66.75), \\ z_{1, 2} & = & (9.0625, 26.5000, 37.5000, 82.500), \\ z_{2, 0} & = & (10.2500, 16.6250, 21.1250, 60.00), \\ z_{2, 1} & = & (10.5000, 20.3750, 28.1250, 66.75), \\ z_{2, 2} & = & (9.2500, 25.6250, 36.1250, 82.500), \\ z_{3, 0} & = & (13.0000, 21.5000, 24.0000, 56.00), \\ z_{3, 1} & = & (13.2500, 25.2500, 31.000, 62.750), \\ z_{3, 2} & = & (12.0000, 30.5000, 39.0000, 78.50), \end{matrix}$

which are given by the Cartesian product of sets points: $\begin{matrix} X & = & {(0.25, 0.5, 1), (0.25, 1, 1.5, 2), \\ (0.5, 0.75, 1.25, 2), 3}, \\ Y & = & {(2, 2.5, 4), (2.25, 3.75, 4.25, 4.75), \\ (1, 5.5, 6.25, 6.5)} . \end{matrix}$

For the above set points, the exact solution function isf (x, y) = (1, 2, 4) x² + (1, 3, 4, 9) y + 8. The bivariate interpolation formula has 12 nodes, and is of degree 3 in x and of degree 2 in y. Here, it will be suggested that four layers feed-forward net is sufficient to estimate the solution polynomial. Now, the incremental learning process is treated to learn the above training pattern for error correction. First, the initial weight parametera_i,jis quantified with small random value to begin the process, which has been chosen on interval [0, 1].

Then the training patterns are used to successively adjust the connection weights until a suitable solution is found. Usually, more than one step through the training set is needed to cause an appropriate solution vector.

The cost function is depicted in Fig. 2 on the 150 iterations. Figure 3 shows the accuracy of the solution polynomial on different values of ®. The obtained numerical results are plotted in Table 1. Comparisons between the proposed algorithm on different training steps and the bivariate cubic spline approximation scheme (BCS) [13], are presented here. It is unequivocally clear that our method can compete in computational efficiency with indicated numerical scheme. All results indicate excellent match between the neural network based solution and the exact one. Also, the accuracy of the obtained results can be improved by growing the number ofiterations.

In practice, the actual number of neural nodes are not arbitrary or random. It is left out as an exercise for the reader to verify the total number of possible neurons in each hidden layers, for realizing such an appropriate fuzzy polynomial. By ascending the number of hidden layers, the proposed method is getting more sufficient to achieve satisfactory accurate results.

Example 4.2. In order to demonstrate how the proposed algorithm can be easily used for functions estimating, a problem similar to the one given in previous example is studied. So, consider the following bivariate fuzzy polynomial: $f (x, y) = x^{3} + (2, 3, 5) x^{2} y + (1, 3, 4) {xy}^{2} + y^{3}$

with its support nodes

x∖y (1, 4, 5) (4, 5, 6) (3, 6, 8) (2,7,8)

(1,3,4) (4,189,640) (9,279,864) (4,387,1120) (9,657,1728)

(2,4,5) (18,316,945) (32,448,1250) (18,604,1595) (32,988,2405)

(1,5,6) (4,485,1320) (9,665,1716) (4,875,2160) (9,1385,3192)

(2,6,7) (18,702,1771) (32,936,2268) (18,1206,2821) (32,1854,4095)

While the initial network parameters are chosen based on assumptions which are seen in previous example, the iterative process yields the results which have been gathered in Table 2. Also, this example is going to show the difference between proposed algorithm and cubic spline approximation method which has been employed in [13]. Similarly, the accuracy of solution polynomials are plotted in Fig. 4.

It follows from the results of these examples that φ (x, y) converges as r, m→ ∞ to the exact solution f (x, y) of the integral equations system. A successful choice of the “zeroth” approximation a_i,j can result in a rapid convergence of the procedure.
5 Conclusion

x∖y	(1, 4, 5)	(4, 5, 6)	(3, 6, 8)	(2,7,8)
(1,3,4)	(4,189,640)	(9,279,864)	(4,387,1120)	(9,657,1728)
(2,4,5)	(18,316,945)	(32,448,1250)	(18,604,1595)	(32,988,2405)
(1,5,6)	(4,485,1320)	(9,665,1716)	(4,875,2160)	(9,1385,3192)
(2,6,7)	(18,702,1771)	(32,936,2268)	(18,1206,2821)	(32,1854,4095)

The problem of solvability of a fuzzy interpolation problem has many theoretical and practical applications in several applied fields. Many existing methods have failed in interpolating fuzzy sets. In other words, a major problem with using other iterative methods is that a lot of terms need to be followed to achieve the goal. Furthermore, the basic architecture developed here requires no initial condition or long phases to find interpolation function. As mentioned through the paper, ANNs known as universal approximator which owns the ability of detecting points need to be interpolated. This type of network has been assigned as adaptive systems and is used in such environments, which constantly change and qualified itself very effective at its intended field. The proposed algorithm can be conveniently implemented in range of field to reduce the production cost and operation times. The majority of practical applications of neural networks currently made us survey a possible iterative algorithm to construct a bivariate fuzzy interpolation polynomial via artificial neural networks approach. To provide this solution, a gradient descent based supervised learning rule of the suggested fuzzy feed-forward neural architecture, was derived for minimizing the defined sum of squared criterion function and gradient descent used to change each weight in proportion to its derivative with respect to the error. Therefore, a global iterative method was discussed by means of ANNs approach which is suited for approximating undetermined fuzzy weight parameters on given training set. In this paper we described models in detail and the various techniques have been explained to train them. Embedding proposed method in a general structure of an ANN has shown the benefit of using available ANN training methods to find the interpolation of arbitrary points. This paper has outlined that feed-forward neural networks are now becoming well established as methods for data processing and interpretation. While feed-forward networks qualifies for the majority of applications there are many other network models, performing a variety of different functions, which we did not have space to discuss in detail here. The simulation results have shown that our method possess the potentiality to behave such an efficient fuzzy rule interpolation technique. It is depicted that by increasing the number of hidden layers, the proposed method will be wound up satisfactory accurate results. The main notable advantage of applying ANNs is that if initial parameters are chosen conveniently, few steps are going to be taken to obtain the exact solutions. Although the discussion here was restricted to the case of two variables functions, it is intended that this review should be accessible to researchers making practical use of techniques which the results can be immediately extended to the multiple cases.

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