金屬材料專業(yè)外文翻譯--利用神經(jīng)網(wǎng)絡(luò)預(yù)測與其他預(yù)測方法對δ鐵素體不銹焊縫的分析和比較（英文）

1、Journal of Materials Processing Technology 142 (2003) 20–28Delta ferrite prediction in stainless steel welds using neural network analysis and comparison with other prediction methodsM. Vasudevan a,?, A.K. Bhaduri a, Bal

2、dev Raj a, K. Prasad Rao ba Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, India b Department of Metallurgy, Indian Institute of Technology, Chennai, IndiaReceived 2 May 2002; receiv

3、ed in revised form 11 December 2002; accepted 17 February 2003AbstractThe ability to predict the delta ferrite content in stainless steel welds is important for many reasons. Depending on the service requirement, manufac

4、turers and consumers often specify delta ferrite content as an alloy specification to ensure that weld contains a desired minimum or maximum ferrite level. Recent research activities have been focused on studying the eff

5、ect of various alloying elements on the delta ferrite content and controlling delta ferrite content by modifying the weld metal compositions. Over the years, a number of methods including constitution diagrams, Function

6、Fit model, Feed-forward Back-propagation neural network model have been put forward for predicting the delta ferrite content in stainless steel welds. Among all the methods, neural network method was reported to be more

7、accurate compared to other methods. A potential risk associated with neural network analysis is over-fitting of the training data. To avoid over-fitting, Mackay has developed a Bayesian framework to control the complexit

8、y of the neural network. Main advantages of this method are that it provides meaningful error-bars for the model predictions and also it is possible to identify automatically the input variables which are important in th

9、e non-linear regression. In the present work, Bayesian neural network (BNN) model for prediction of delta ferrite content in stainless steel weld has been developed. The effect of varying concentration of the elements on

10、 the delta ferrite content has been quantified for Type 309 austenitic stainless steel and the duplex stainless steel alloy 2205. The BNN model is found to be more accurate compared to that of the other methods for predi

11、cting delta ferrite content in stainless steel welds. © 2003 Elsevier Science B.V. All rights reserved.Keywords: Neural network analysis; Delta ferrite content; Austenitic stainless steel; Duplex stainless steel1. I

12、ntroductionThe ability to estimate the delta ferrite content accurately has proven very useful in predicting the various properties of austenitic SS welds. A minimum delta ferrite content is necessary to ensure hot crack

13、ing resistance in these welds [1,2], while an upper limit on the delta ferrite content de- termines the propensity to embrittlement due to secondary phases, e.g. sigma phase, etc., formed during elevated tem- perature se

14、rvice [3]. At cryogenic temperatures, the tough- ness of the austenitic SS welds is strongly influenced by the delta ferrite content [4]. In duplex stainless steel weld metals, a lower ferrite limit is specified for stre

15、ss corrosion cracking resistance while the upper limit is specified to ensure ade- quate ductility and toughness [5]. Hence, depending on the service requirement a lower limit and/or an upper limit on delta ferrite conte

16、nt is generally specified. During the selec-? Corresponding author. Tel.: +91-4114-80232; fax: +91-4114-40381. E-mail address: dev@igcar.ernet.in (M. Vasudevan).tion of filler metal composition, the most accurate diagram

17、 to date WRC-1992 is used generally to estimate the ?-ferrite content [6]. The Creq and Nieq formulae used for generat- ing the WRC-1992 constitution diagram is given by Creq = Cr+Mo+0.7Nb and Nieq = Ni+35C+20N+0.25Cu. T

18、he limitation of these equations is that values of the coefficients for the different elements remain unchanged irrespective of the change in the base composition of the weld. However, the relative influence of each allo

19、ying addition given by the elemental coefficients in the Creq and Nieq expressions is likely to change over the full composition range. Further- more, these expressions ignore the interaction between the elements. Also,

20、there are a number of alloying elements that have not been considered in the WRC-1992 diagram. Ele- ments like Si, Ti, W have not been given due to consider- ations, though they are known to influence the delta ferrite c

21、ontent. Hence, the delta ferrite content estimated using the WRC-1992 diagram would always be less accurate and may never be close to the actual measured value. In the Function Fit model [7] for estimating ferrite, the d

22、ifference in free energy between the ferrite and the austenite was calculated0924-0136/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00430-822 M. Vasudevan et al. /

23、 Journal of Materials Processing Technology 142 (2003) 20–28Fig. 1. Schematic diagram of the network structure showing the input nodes, hidden units and the output node.function so that each input contributes to every hi

24、dden unit, where N is the total number of input variables:hi = tanh??N ?j w(1) ij xj + θ(1) i?? (3)Here the bias is designated as θ and is analogous to the constant in linear regression. The transfer from the hidden unit

25、s to the output is linear, and is given by:y =N ?i w(2) i hi + θ(2) (4)The output y is therefore a non-linear function of xj, with the function usually selected for flexibility being the hyperbolic tangent. Thus, the net

26、work is completely described if the number of input nodes, output nodes and the hidden units are known along with all the weights wij and biases θi. These weights wij are determined by training the network and involves m

27、inimization of a regularized sum of squared errors. The BNN analysis of Mackay [10] allows the calcula- tion of error-bars with two components—one representing the perceived level of noise (σv) in the output and the othe

28、r indicating the uncertainty in the data fitting. This latter com- ponent of the error-bars emanating from the Bayesian frame- work allows the relative probabilities of the models with different complexity to be assessed

29、. This enables estimation of quantitative error-bars, which vary with the position in the input space depending on the uncertainty in fitting the func- tion in that space. Hence, instead of calculating a unique set of we

30、ights, a probability distribution of weights is used to define the uncertainty in fitting. Therefore, these error-bars become large when data are sparse or locally noisy. In this context, a very useful measure of the err

31、or is the logarithm of the predictive error (LPE) given by the following:LPE = ?n12? (t(n) ? y(n))2σ(n)2 y + log(2πσn y)1/2 ?(5)where t is the target for the set of inputs x, while y the cor- responding network output. σ

32、y is related to the uncertainty of fitting for the set of inputs x. By using LPE, the penaltyfor making a wild prediction is reduced if that prediction is accompanied by an appropriately large error-bar, with a larger va

33、lue of the LPE implying a better model. Further, by this method it is also possible to automatically identify the input variables that are significant in influencing the output variable, as the input variables that are l

34、ess significant are down-weighted in the regression analysis.3.1. Over-fitting problemIn BNN analysis, two solutions are implemented which contribute to avoid over-fitting. The first is contained in the algorithm due to

35、MacKay [12]: the complexity parameters α and β are inferred from the data, therefore allowing auto- matic control of the model complexity. The second resides in the training method. The database is equally divided into a

36、 training set and a testing set. To build a model, about 80 networks are trained with different number of hidden units and seeds, using the training set; they are then used to make predictions on the unseen testing set a

37、nd are ranked by LPE.3.2. Committee modelThe networks with different number of hidden units will give different predictions. But predictions will also depend on the initial guess made for the probability distribution of

38、the weights (the prior). Optimum predictions are often made using more than one model, by building a committee. The prediction ¯ y of a committee of networks is the average prediction of its members, and the associa

39、ted error-bar is calculated according to Eq. (6):¯ y = 1L?l y(l)σ2 = 1L?l σ(l)2 y + 1L?l (y(l) ? ¯ y)2 (6)where L is the number of networks in a committee. Note that we now consider the predictions for a given