机械设计方面的外文参考文献

Set of NN weights w!p) li=1, ... ,NW;p=l, ... ,L
Stage III Computation of membership functions for NN weights
FuzyNN with weights membership functions Pi = p(w~p»
inequalities (l-2KIL)::;; IX < (l-2(K-I)/L , where: K = kLa . Ita - numbers of
weight values on the left or right hand sides of w, respectively. In case of a E
There are three possibilities to formulating fuzzy networks. The first one corresponds to the neural network with crisp parameters (called for short NN weights) and performing computations on interval variables [8]. Much advanced are NNs with crisp inputs and outputs but their processing is performed on fuzzyfied variables with fuzzy reasoning rules, cf. fuzzy inference systems [4]. The third class is associated with full fuzzification of transmitted signals, NN weights and neurons of a fuzzy NN [2]. A more numerically efficient approach depends on joining simple membership functions of signals and NN parameters with interval arithmetics [7].
Fig. 1. Schematic algors assume that the network was designed using a corresponding crossvalidation procedure and a subset selected from (1). The formulated network is then trained on set (1) at Stage I of the algorithm shown in Fig. 1. A set ofNN weights (both synaJ?tic weights and biases) is collected as a set of initial value
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w, respectively. The interval values of a-cut [wL , wR]a are depicted in Fig. 2a
as WL",WR".
a)
p(w)
1.0
b)
Jt(W)
1.0
(t) a = 0.,5
Wmin W{;s ill w{fs
W W max
w, mited). The distances 3 aL and 3aR are measured from the mean value
where: aL, aR- standard deviations of patterns p that are smaller or greater than
weights Wo= {wed i= 1, ... ,NW}, where: NW-numberofNN weights. The weights WOi are assumed as initial weights to learn weights correspond-
ing to each pattern of the training set (l). At Stage II the network is trained L times for a sequence of single patterns p = 1,... ,L. After this training a set of
Ii weights is completed, i.e. W= {Wi }= [Wi (P) = 1, ... ,NW; p = 1,... , L] .
The membership functions for the NN weights J1 i = J1 (W;) are computed at Stage III. In [6] the triangular membership functions were assumed. In Fig.2a a triangular shape of MB function (t) is shown for the weight w (index i is om-
Fig. 2. Shapes of membership functions for NN weights: a) triangular MF (t), b) nonlinear MF (n)
The other method of formulation of a nonlinear MF (function (n) in Fig. 2b)
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gorithm is shown. It corresponds to a standard multi-layered, forward neural network and the training set of patterns:
I £ = {(x,t)(p) p= 1,...,L}
(1)
1. Introduction
Results of tests on material models can be noisy, incomplete and inconsistent. Another aspect is a limited number of tests because of various difficulties and costs of arrangement of experiments on laboratory specimens or full scale structures. That is why fuzzy variables and a possibility approach seem to match better the nature of experimental results than using crisp variables [5] . This concerns also fuzzy neural networks.
Set of training patterns
.c = {(x, t)(p) I p = 1, ... ,L}
Stage I Initial training ofNN
Initial values ofNN weights
{w? Ii == 1, ... ,NW}
Stage II Detail training of NN
Crisp type neurons are used for the transformation of interval values for fixed a
- cuts, transmitted through NN. The above mentioned approach is developed in the paper. The correspond-
In the paper a simplified approach, proposed in [6] is discussed. The main idea lies in the formulation of NN weights on the basis of weight values, computed for single patterns taken each after the other from the training pattern set.
( K- 1 , K) the values wLa , wRa are computed by linear interpolation in the range (~-1, wK ).
After the membership functions are formulated for each NN weight the
A Fuzzy Neural Network for the Analysis of Experimental Structural Mechanics Problems
Ewa Pabisek, Magdalena Jakubek and Zenon Waszczyszyn
Institute of Computer Methods in Civil Engineering, Cracow University of Technology, Warszawska 24, 31-155 Krak6w, Poland, e-mail: zenwa@.pl
The idea of a fuzzy NN, proposed in [6], is shown Fig. I where a schematic al-
L. Rutkowski et al. (eds.), Neural Networks and Soft Computing © Springer-Verlag Berlin Heidelberg 2003