**9. References**

20 Recurrent Neural Networks and Soft Computing

**0 50000 100000 150000 2**

Fig. 10. The optimisation process seeks the neural network that uses both the lowest possible number of epochs while producing an approximation with the narrowest training stability band. Each dot on the graph represents one neural network configuration. Interesting

The field of neural networks as a tool for approximation (modelling) is very rapidly

In this chapter, two very important issues of the theory of approximation are questioned. One is the universality of approximation, and the second is the best approximation property. Both properties are not applicable to the neural networks that run on digital

Theoretically, it has been proven that a three layer neural network can approximate any kind of function. Although this is theoretically true, it is not a practical situation since the number of epochs needed to reach the prescribed approximation precision drops

The newly introduced concept of approximation of wide-range functions by using logarithmic or segmented values gives the possibility to use neural networks as approximators in special cases where the modelled function has values spanning over

The training stability analysis is a tool for assessment of training diversity of neural networks. It gives information on the possible outcomes of the training process. It also

evolving. Numerous authors have reported their research results on the topic.

significantly with the increase in the number of hidden layers.

provides the ground for further optimization.

**1 5 15 20 1**

**1 20 5 5 1**

**average nu m ber of epo ch s**

**1 40 1**

**0**

**1 15 20 15 1**

**1 20 20 20 1**

configurations are highlighted.

**8. Conclusions** 

computers.

several decades.

**0 .0 0 2**

**0 .0 0 4**

**0 .0 0 6**

**0 .0 0 8**

**average stability band width**

**0.01**

**0 .0 1 2**

**0 .0 1 4**

**0 .0 1 6**

**0 .0 1 8**


**A Framework for Bridging the Gap Between** 

*2Computer Science and Eng. Dept, Egypt-Japan University of Science and Technology* 

FRS (Fuzzy Rule Systems) and ANNs (Artificial Neural Networks) have gained much popularity due to their capabilities in modeling human knowledge and in learning from data, respectively. Fuzzy systems have the advantage of allowing users to incorporate available experts' knowledge directly in the fuzzy model [2]. Thus, decisions made by fuzzy systems are transparent to the user (i.e. the reasons behind the decisions made are clearly understood by tracing the decision and finding out which rules fired and contributed to it). However, there are many parameters whose values are arbitrary. These values have to be "guessed" by a fuzzy system designer, yet they largely influence the system behavior. Thus,

On the other hand, ANNs have the advantage of being universal functions approximators, requiring only sample data points and no expert knowledge [3]. Despite their advantages, they are essentially black box models. This means that the reasons behind their decisions are concealed in the knowledge acquired in the trained weights. However, these usually have

To combine the advantages of both systems while overcoming their disadvantages, two approaches have been proposed in literature. The first is rule extraction from weights of trained ANNs [4]-[7]. However, the proposed approaches often yield some "un-plausible" rules, thus rule pruning and retraining is often required. For examples, some rules may be impossible i.e. their firing depends on conditions that can never occur in reality (*impossible antecedents*). For example, a rule dictating that a certain action is to be taken in case time is negative. The second approach is ANFIS [8], which attempts to cast the fuzzy system as an ANN with five layers. Although, only two of these layers are adaptable, this model is still more complicated to build and train than a conventional feed-forward ANN for two main reasons. First, the user's expertise is required to choose appropriate consequent (output) membership functions. Second, the desired output needs to be known a priori. This may not be possible for several applications including design problems, inverse problems and high dimensional problems. For example, in a robot path tracking problem, the ANN is required to predict the correct control input. In such application, the desired performance is known but no real-solid rules exist, especially, if the robot is required to be self-adaptive. Similarly,

**1. Introduction** 

a fuzzy system is as good as its programmer.

no clear logical interpretation. Thus, their reliability is questionable.

**Symbolic and Non-Symbolic AI** 

Gehan Abouelseoud1 and Amin Shoukry2

*1Alexandria University* 

*(EJUST), Alexandria,* 

*Egypt* 

Wray, J., Green, G.G.R. (1994). Neural Networks, Approximation Theory and Finite Precision Computing. *Neural Networks*. 8 (1) **2** 
