Fuzzy neural networks and neural fuzzy systems are powerful tools for computational, control, and predictive applications. Basic ideas are presented here to allow the reader to pursue the subject with further research. Assumptions are made that the reader understands the basics of fuzzy logic, neural networks, perceptrons, and the basic Kohonen (VQ) network.

 

In basic examples of neural networks such as the perceptron network, Kohonen self-organizing network, or Adaptive Resonance Theory (ART) networks, the models are feed forward. When neurons in a feed forward network compete only the winner learns. The losing neurons do not learn. These models are not efficient in using the neurons to their full potential. It is even possible that certain neurons will go unused if their weights are not close to what is required from the training data.  In such networks the inputs are crisp, while a convergence process finalizes the weights into fixed form by repeating the algorithm:

 

w_ij(t + 1) = w_ij(t) + h(t)(x_i(t) w_ij(t)),

 

where j Î neighborhood size [NE(t)],  (n-1) ³ i ³ 0, and h(t) is the gain factor that represents the learning rate (0 < h(t) < 1) that decreases over time.

 

Once the term h(t) is reduced to zero the weights become fixed. Poor/incorrect data may result from this set-up if the network converges on a local minimum instead of a global minimum from the data set.

 

By using fuzzy competitive learning we can overcome these handicaps of traditional competitive learning in feed forward networks. Instead of each node being a winner or loser, they are all given membership to the set of winners via a fuzzy scaling factor. This will result in degrees of winners versus one dominant winner. The neurons are then able to learn based upon their membership to the winners set. To perform this we modify the weight distribution algorithm previously discussed to:

 

w_ij(t + 1) = w_ij(t) + h(t)Z_ij(t)[x_i w_ij(t)],

 

Where we introduce Z_ij as the scaling factor that is a function of the membership m_j, which is assigned to each neuron according to its computed distance between the output vector and weight vector. Again once the values are initialized, the process is repeated until h(t) is reduced to zero thereby converging the weights to their fixed value.

 

We just applied fuzzy logic to the weights of a Kohonen network with crisp inputs to modify the output. It is also possible to apply fuzzy concepts to other aspects of neural networks. Pal and Mitra worked on two examples of this [1,2].  They were able to apply fuzziness to the input of a network to eliminate corruption of data (noise for example). They accomplished this by applying a hedge (terms that intensify, dilute, or complement a fuzzy set) to associate linguistic values (very low, low, medium, high, very high) instead of empirical data. Also they were able to apply fuzziness to the back-propagation (supervised learning rule that calculates the error function for a known input then propagates the error backwards through the network to achieve a stable state) of a multi-layered perceptron network. A linguistic hedge was applied to the inputs of the network. During back-propagation the errors in membership were calculated with an assigned membership function giving certain nodes more weight. While the ending results are favorable, the network requires supervised learning (training) and testing, which is very time consuming.

 

As you can see there are several possible ways for fuzzy logic and neural networks to interact. There are not only several methods but also several models, with many more to come. Please review the references for more information on fuzzy logic, neural networks, and their relationship.

 

 

 

Reference:

1. S.K. Pal and S. Mitra, Multilayer Perceptron, Fuzzy Sets, and Classification, IEEE Trans. Neural Networks 3 (1992): 683-697
2. S. Mitra and S.K. Pal, Self Organizing Neural Networks as a Fuzzy Classifier, IEEE Trans. Systems, Man. And Cyber, 24 (1994): 385-399
3. For good books on Neural Networks, take a look at
   ftp://ftp.sas.com/pub/neural/FAQ4.html
4. comp.ai.neural-nets' FAQ
   at ftp://ftp.sas.com/pub/neural/FAQ.html
5. Ahamad M. Ibrahim, Introduction to Applied Fuzzy Electronics (Prentice Hall, 1997)
6.  Fuzzy FAQ  www-2.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html