COMPARISON OF USAGE OF DIFFERENT NEURAL STRUCTURES TO PREDICT AAO LAYER THICKNESS

Original scientific paper The paper deals with the comparison of usage of three basic types of neural units in order to create the most suitable model predicting determining the final thickness of the alumina layer formed at surface with current density of 1 A∙dm. In addition, the reliability of obtained prediction models, depending on the amount of training data, has been monitored. With properly selected range of training data it is possible to create prediction models with reliability greater than 95 % with achieved toleration 2×10 mm.


Introduction
Pure aluminium and its alloys, as weight-reducing materials, are becoming more significant not only from technical, but also from technological and economic standpoint [1], [2] mostly in aerospace and automobile industries, where light and sturdy structures are preferred [3], [4].Using these materials for moving parts presents a great challenge when high resistance to abrasion and wear is required [5].These tribological properties can be improved by anodic oxidation of components surface.Anodizing is one of the most important processes in corrosion protection and colour finishes for aluminium [6].Design of experiments (DoE) is one of the basic tools which help us to show the influence of input factors on outputs [7][8][9][10].The optimum selection of process conditions is an extremely important issue as these determine surface quality of the manufactured components [11][12][13].The mathematical modelling of the process involves several parameters that may lead to difficult analytical solution [14], [15].On the other hand, use of artificial intelligence for evaluation of experiments results has its merits mainly because of faster and more reliable creation of prediction model for the studied process, compared to classic statistical methods [16][17][18][19].

Methods
Based on DoE, we have oxidized 46 samples of alloy EN AW 1050A.Chemical composition of electrolytes was in accordance with central composite design of experiment.Composition of electrolyte was influenced by amount of sulphuric acid in electrolyte (factor x 1 ), amount of oxalic acid in electrolyte (factor x 2 ) and amount of aluminium cations in electrolyte (factor x 3 ).Operating conditions were: electrolyte temperature (factor x 4 ), time of oxidation (factor x 5 ) and applied voltage level (factor x 6 ).Naming of observed factors and their levels in natural and coded scale are shown in Tab. 1.We have measured the thickness of aluminium oxide layer on each sample after the oxidation.Layer thickness was measured in the area with surface current density 1 A•dm −2 .
We have used three types of neural units to compile prediction model: 1 st order HONU (Higher-Order Neural Unit) (linear neuron unit), 2 nd order HONU (quadratic neural unit) and 3 rd order HONU (cubic neuron unit).These neural units compiled the prediction model based on adaptive optimization algorithm -Levenberg-Marquardt [20], [21].
This algorithm is described by Eq. ( 1) ÷ (8).It is a process of updating individual weights w in a predetermined number of steps to achieve a minimum difference between the real (measured) and calculated values.The vector u of neural inputs is created by taking the partial derivatives of each output in respect to each weight Eq. (1) -Eq.(3).The equation describing the investigated model is the characteristic equation of given type of neural unit (1 st order HONU, 2 nd order HONU a 3 rd order HONU) for observed factors x 1 , x 2 , x 3 , x 4 , x 5 , x 6 [22], [23].Technical Gazette 24, 2(2017), 333-339 The Levenberg-Marquardt algorithm consists of solving Eq. ( 4), where the Jacobian J is the matrix of dimension n×m , where n is the length of the input vector u of the neural unit (n is the number of neural inputs) and m is the total number of parameters intended for the learning procedure.
The vector y' of neural outputs is defined as the dot product of vectors w and u Eq. ( 5), the size of the individual weight is set in the first step randomly.
After calculating the output vector, the error vector e is calculated as the difference between the actual value y of the observed variable and the calculated one by the neural units y', as seen in Eq. (6).
Then the weight update vector Δw is determined by Eq. (7).In Eq. ( 7) is the weight update vector Δw that we want to find, e is the error vector containing the output errors for each input vector used on training the network, 1/μ is the Levenberg's damping factor which tells us by how much we should change our network weights to achieve a (possibly) better solution.The J T •J matrix can also be known as the approximated Hessian, the I is an identity matrix of diagonal length equal to the number of neural weights (matrix of dimension n×n), and μ is the learning rate.The size of the learning rate μ depends on how quickly and how accurately the neural unit is able to learn.
After calculating the weight-updates Eq. ( 8), the adaptation of the weights of input factors occurs.Weight update vector Δw is sum up to actual vector of weight w.The learning process of neural units continues by calculating the vector of neural outputs y using the new (adapted) weights.

Δw
After the learning process of neuron unit is done, we get a computational model that describes the thickness of AAO layer with equation Eq. ( 9) and Eq. ( 10) for linear neural unit, Eq. ( 11) and Eq. ( 12) for quadratic neural unit and Eq. ( 13) and Eq. ( 14) for cubic neural unit.
where th is final thickness of oxide layer, α is preliminary thickness of oxide layer, u i is a combination of input factors levels (in coded scale), w i are weights for combinations of input factors and std y is standard deviation of real values (measured layer thickness vector y) divided by 3. Calculated thickness of oxide layer is expressed in 10 −3 mm.

Results
We have gradually reduced range of training data during the neural unit's learning process.At first, we have used all of the 46 measurements of AAO (anodic aluminium oxide) layer thickness.The number of AAO layer thickness values was reduced by one with each subsequent iteration of the learning process.The lowest range of training data was 10 measurements of thickness.Tab. 2 shows learning error (sum of square errors for learning process for linear neural unit -sse LLNU , quadratic neural unit -sse LQNU and cubic neural unit -sse LCNU ) for each neural unit in selected range of training data (number of training values -NoTV).
From shown sums of squared errors (Tab.2), we can assume that each type of neural unit can produce more reliable prediction model with less training data.
However, process of verification denies this assumption.
Figs. 1 ÷ 3 show an error of predicted AAO layer thickness for linear, quadratic and cubic neural unit.Each neural unit used all of the 46 measurements of AAO layer thickness during the learning process.Linear prediction model created by LNU (Fig. 1) is comparatively accurate because error of prediction is less than ±4,00×10 −3 mm for 39 samples.On the other hand error of prediction is greater than 4,00×10 −3 mm for 7 samples, especially for sample of number 33, where the prediction error is 14,50×10 −3 mm.Therefore, it is possible to use this prediction model only as indication to the AAO layer thickness.Use of this model for optimization of aluminium anodizing is impossible.Fig. 2 shows that QNU was able to create a model with high prediction capability.We can see in Fig. 2 that 37 samples are in toleration area ±2,00×10 −3 mm.The other 6 samples are in toleration area ±3,00×10 −3 mm and finally 3 samples are in toleration area ±4,00×10 −3 mm.Therefore, it is possible to use this quadratic prediction model for optimization of anodic oxidation of aluminium but only under certain conditions.For example if the toleration area ±5,00×10 −3 mm is acceptable.If we use all measured layer thickness measurements in learning process it can be very difficult to determine if the prediction model is compiled correctly, because we do not have any measurements to validate the compiled prediction model.There is a risk of overtraining the neural unit.This is the reason why we have gradually reduced range of training data during the learning process.
Tab. 3 shows the sum of squared errors between measured and calculated values of AAO layer thickness for each neural unit.Generally it can be concluded from Tab. 3 that with decrease in amount of training data also Technical Gazette 24, 2(2017), 333-339 decreases the reliability of prediction of AAO layer thickness, but there are some exceptions.The sum of squared errors for CNU is increasing from 0,8 (46 training data are used) to 59 (43 training data are used).Then it starts decreasing to 8,6 (36 training data are used).After that it is still increasing.
From Tab. 3 we can conclude that examined neural unit can compile very accurate prediction model if the sum of squared errors is ranging from 0,00 to 50,00.Prediction model is considered accurate if the sum of squared errors is ranging from 50,00 to 200,00.Prediction model can be used only for approximation if the sum of squared errors is ranging from 200,00 to 500,00.Prediction model is considered unusable with sum of squared errors greater than 500,00.Fig. 4, Fig. 5 and Fig. 6 show an error between computed and measured AAO layer thickness for linear, quadratic and cubic neural units, which used 36 values of AAO layer thickness during their learning process.Linear prediction model (Fig. 4) is still usable for approximate calculations.Error of prediction is less than ±4,00×10 −3 mm for 37 samples and error of prediction is greater than 4,00×10 −3 mm for 9 samples.Fig. 5 shows that error for quadratic prediction model is too large (almost 45,00×10 −3 mm) for the model to be usable.Its accuracy is further reduced with a decrease in amount of training data.Cubic prediction model (Fig. 6) is still very reliable.Error of prediction is in toleration area ±1,50×10 −3 mm for 3 samples, in toleration area ±0,50×10 −3 mm for 4 samples and error of prediction is 0,00 mm for other 36 samples.9 show a comparison of measured and predicted AAO layer thickness for LNU, QNU and CNU, which used 28 values of AAO layer thickness during their learning process.As we can see from Fig. 7 LNU still has capability to compile an approximate predictive model.Error of prediction is usual in range ±6,00×10 −3 mm, but it is mostly in range ±2,00×10 −3 mm.The prediction model compiled by QNU which use 28 training data (Fig. 8) is unusable the same as in previous case.As shown in Fig. 8, error for quadratic prediction model is too large (almost 25,00×10 −3 mm) for the model to be usable.Its accuracy is further reduced with a decrease in amount of training data.As shown in Fig. 9, cubic prediction model compiled by CNU is still very precision again Every error of prediction is in toleration area ±2,00×10 −3 mm.But this is last time when the cubic prediction model is usable to optimize the examined process of anodizing.Fig. 10, Fig. 11 and Fig. 12 show error of prediction for LNU, QNU and CNU.As we can see from Fig. 10 linear prediction model is relatively inaccurate because error of prediction is in a very wide range (from −8,00×10 −3 to 6,00×10 −3 mm).This is a problem because 8,00×10 −3 mm of layer thickness represents 5 to 10 years of corrosion protection (although this depends on environmental influence).This is the reason why we cannot use this prediction model even to approximate calculations.

Conclusion
Main objective of presented study was to create a prediction model which is able to reliably predict the resulting thickness of aluminium oxide layer, based on factors which enter the technological process of anodic aluminium oxidation.This paper shows that use of higher-order neural units has great potential for evaluation of experiments results.With properly selected range of training data, it is possible to create prediction models with reliability greater than 95 %.Such high reliability offers possibilities for off-line optimization of examined production processes.The usage of developed prediction model allows us to reduce the operating cost and simultaneously create desired value of AAO layer thickness.Finally we can state that it is possible to increase the corrosion resistance of treated components and extend their lifetime.However, use of presented algorithm also brings with it 2 significant problems: 1.It is necessary to create a preliminary computational model on which neural unit will base its learning process and subsequently control it.2. Resulting computational model is not necessarily able to sufficiently describe controlled process and thus increases the probability for failed product.
More research should expand the prediction model by adding larger spectrum of current densities and also more precisely define influence of physical parameters on resulting AAO layer thickness.

Figure 1
Figure 1 Validation of LNU for training process with 46 training data

Figure 2
Figure 2 Validation of QNU for training process with 46 training data Fig. 3 shows that CNU was able to create almost perfect prediction model.It can be seen that only 2 samples have error of computation ±0,60×10 −3 mm.All other samples have error of computation less than 0,20×10 −3 mm (mostly less than 0,05×10 −3 mm).It means that this CNU prediction model can be considered accurate.So it is possible to use this model to optimize process of aluminium anodizing.

Figure 3
Figure 3 Validation of CNU for training process with 46 training data

QNU
has smaller error of prediction than LNU only if we use from 46 to 41 training values.Then the sum of square errors is rapidly increasing with each decrease of training data for QNU, until we reach 25 training data.After that the sum of squared errors for QNU starts decreasing again, until we reach 15 training data.Then it starts increasing.Sum of square errors for LNU is relatively at constant level all the time.It gradually increases from 480 to 670 until we reached only 15 training data.If we use 10 or 11 training data for LNU, the prediction model is more precise than if we use more training data (12, 13 or 14).

Figure 4 Figure 5
Figure 4 Validation of LNU for training process with 36 training data

Figure 6
Figure 6 Validation of CNU for training process with 36 training data

Figure 7 Figure 8 Figure 9
Figure 7 Validation of LNU for training process with 28 training data

Figure 10 Figure 11
Figure 10 Validation of LNU for training process with 23 training data

Figure 12
Figure 12 Validation of CNU for training process with 23 training data Tab. 4 shows sum of squared errors between measured and calculated values of AAO layer thickness for each neural unit.From Tab. 4 we can conclude that examined cubic neural unit can compile very accurate prediction model if we use at least 28 training data.Cubic prediction model is considered accurate if we use at least 23 training data and prediction model can be used only for approximation if we use at least 16 training data.

Table 1
Factors level in nature and coded scale

Table 2
Learning error of neural units

Table 3
Sum of square errors for linear, quadratic and cubic neural unit

Table 4
Adjustment of prediction models compiled by LNU, QNU and CNU