A novel technique for solving fully fuzzy nonlinear systems based on neural networks

Predicting the solutions of complex systems is a crucial challenge. Complexity exists because of the uncertainty as well as nonlinearity. The nonlinearity in complex systems makes uncertainty irreducible in several cases. In this paper, two new approaches based on neural networks are proposed in order to find the estimated solutions of the fully fuzzy nonlinear system (FFNS). For obtaining the estimated solutions, a gradient descent algorithm is proposed in order to train the proposed networks. An example is proposed in order to show the efficiency of the considered approaches.

The fuzzy variable c is demonstrated as in which c is the lower-bound variable and c is the upper-bound variable.
De¯nition 2. The Z-number is made up of two components Z ¼ ½cð&Þ; p. The¯rst component cð&Þ is the restriction on a real-valued uncertain variable &. The second component p is a measure of reliability of c. p can be reliability, strength of belief, probability or possibility. The Z-number can be stated as Z þ -number in the case where cð&Þ is a fuzzy number and p is the probability distribution of &. If cð&Þ as well as p are fuzzy numbers, then the Z-number can be stated as Z À -number.
The Z þ -number contains more information when compared with the Z À -number. In this paper, the de¯nition of Z þ -number is utilized, i.e. Z ¼ ½c; p, where c is a fuzzy number and p is a probability distribution. The most common membership functions which de¯ne the fuzzy numbers are the triangular function and the trapezoidal function The probability measure can be stated as where p is the probability density of & and < is the restriction on p. For discrete Z-numbers we have The -level for fuzzy number c is stated as where 0 < 1, c 2 E.
Therefore ½c 0 ¼ c þ ¼ f& 2 <; cð&Þ > 0g. As 2 ½0; 1, ½c is bounded, c ½c c . The -level of c between c and c can be de¯ned as where c as well as c are functions of . We de¯ne c ¼ d A ðÞ, c ¼ d B ðÞ, 2 ½0; 1.

De¯nition 4.
The -level of the Z-number Z ¼ ðc; pÞ is de¯ned as follows: where 0 < 1. ½p is computed by Nguyen's theorem, where pð½c Þ ¼ fpð&Þj& 2 ½c g. Therefore, ½Z is stated as where P ¼ c pð& Similar to the fuzzy numbers, 1 three main operations are de¯ned for the Z-numbers: È; É and , which indicate the sum, subtract and multiply, respectively. In this paper, the proposed operations are di®erent from the ones in Ref. 33.
Suppose Z 1 ¼ ðc 1 ; p 1 Þ as well as Z 2 ¼ ðc 2 ; p 2 Þ are two discrete Z-numbers expressing the uncertain variables & 1 and & 2 , so P k¼1 p 1 ð& 1k Þ ¼ 1; The following operation is de¯ned: where ÃfÈ; É; g. The operations used for the fuzzy numbers are stated as 1 For the discrete probability distributions, the following relation is de¯ned for all p 1 Ã p 2 operations: The Hukuhara di®erence is de¯ned as 35 In case that Z 1 É H Z 2 exists, the -level can be de¯ned as Moreover, the generalized Hukuhara di®erence is de¯ned as 36 By taking into consideration the -level, we have ( Suppose c is a triangular function, the absolute value of the Z-number Z ¼ ðc; pÞ is de¯ned as Let c 1 as well as c 2 be triangular functions, the supremum metric for the Z-numbers Z 1 ¼ ðc 1 ; p 1 Þ and Z 2 ¼ ðc 2 ; p 2 Þ is expressed as where dðÁ; ÁÞ is the supremum metric for fuzzy sets. 1 DðZ 1 ; Z 2 Þ has the properties mentioned in the following: where b 2 <, Z ¼ ðc; pÞ is the Z-number and c is a triangle function.
Using the de¯nition of generalized Hukuhara di®erence, the gH-derivative of H at c 0 is de¯ned as In (22), Hðc 0 þ Þ as well as Hðc 0 Þ represent symmetric pattern with Z 1 and Z 2 , respectively, given in (16).

Neural Network Approach for Z-Number Solution Approximation
Here two novel techniques based on neural networks are suggested for obtaining the numerical solutions of FFNS.

Fully fuzzy nonlinear systems
Let us take into consideration the following system: where S 1j ; S 2j ; ; ; G 1 ; G 2 belong to Z-number set (for j ¼ 1; . . . ; nÞ. For obtaining the approximated solutions, a feedback neural network is developed. The suggested neural network is demonstrated in Fig. 1.

Calculation of Z-number output
Here a feedback neural network is suggested such that the -level sets of the Z-number parameters S qj are considered to be nonnegative, i.e. 0 S qj S qj where j ¼ 1; . . . ; n and q ¼ 1; 2. The following relations are generated: . input units: . hidden units in the¯rst layer: where . hidden units in the second layer: where and . output unit: where A cost function for -level sets of the Z-number output È q and the corresponding target output G q is stated as follows: where is considered to be the learning rate and is considered to be the momentum term constant. @e q @ r is calculated as follows: Hence, @e where @ È q @ r ¼ X j"C @ È q @o j @o j @u j @u j @ð Þ j @ðÞ j @ r þ X j"C @ È q @o j @o j @u j @u j @ð Þ j @ð Þ j @ r ; ð41Þ @ð Þ j @ r as well as @ð Þ j @ r are calculated as follows: and @ð Þ j The connection weights j are updated as follows: We can adjust the Z-number parameter like . The solution of (23) can also be obtained with another type of neural network, see Fig. 2. In this neural network, the -level sets of the fuzzy input S qj are nonnegative, i.e. 0 S qj S qj , where j ¼ 1; . . . ; n and q ¼ 1; 2. The following relations are generated: . input units: . hidden units: where where C ¼ fjj ! 0g, D ¼ fjj < 0, j is eveng and F ¼ fjj < 0; j is oddg, and where C 0 ¼ fjj ! 0g, D 0 ¼ fjj < 0; j is eveng and F 0 ¼ fjj < 0; j is oddg; . output unit: where where A ¼ fjjo qj ! 0; ! 0g, B ¼ fjjo qj ! 0; < 0; j is eveng, H ¼ fjjo qj ! 0; < 0; j is oddg, K ¼ fjjo qj < 0; ! 0g, L ¼ fjjo qj < 0; < 0; j is eveng and T ¼ fjjo qj < 0; < 0; j is oddg, and < 0; j is eveng and T 0 ¼ fjjo qj < 0; < 0; j is oddg. For this neural network the training algorithm is similar to (36).

Numerical Example
In this section, a numerical example is used to demonstrate how to apply feedback neural network and feed-forward neural network in order to¯nd the solutions of FFNS.
Example. Consider the following FFNS:  Tables 1 and 2 for neural networks shown in Fig. 1 (feedback neural network) and Fig. 2 (feed-forward neural network), respectively. It can be seen that feed-forward neural network method and feedback neural network method can approximate the Z-number solutions. Feedback neural network method is more suitable than the feed-forward neural network method. The reason behind it is that the approximation error of feedback neural network can be lower than that of feedforward neural network, while the former needs lower number of training iterations. In Tables 1 and 2, 'ðkÞ and ðkÞ are approximate solutions of FFNS (52), k is the number of iterations and p is the measure of probability. The error between the approximate solution and the exact solution for both approaches is demonstrated in Fig. 3. In Fig. 3 the approximation error of the feedback neural network method is smaller than the feed-forward neural network method.

Conclusion
In this paper, two approaches on the basis of neural networks are suggested for obtaining the estimated Z-number solutions of FFNS. A learning algorithm based on the gradient descent technique is used in order to generate the estimated solutions of FFNS. An example is proposed in order to show the e±ciency of the proposed approaches. The simulation results show that these new models are e®ective to estimate the Z-number solutions of FFNS. The comparison of the feedback neural network method with the feed-forward neural network method shows that the feedback neural network method is better or at least more suitable than the feedforward neural network method. The reason behind it is that the approximation error of feedback neural network can be lower than that of the feed-forward neural network, while the former needs lower number of training iterations. Further work is to study the stability of training algorithm.