The artificial neural network used for the boriding process study is composed of three layers, an input layer made up of two neurons (temperature and boriding time), a hidden layer with five neurons and an output layer representing the thickness of borided layer.
The result obtained by the output layer is represented by the following equation:
where, ?_ij^k : represents the bias, w_ij^k : the weight between neuron i and j , k the layer and g : the activation function.
In this type of simulation, we have three important parameters. At Input, we use the treatment time and the boriding temperature. At output, we obtain the thickness of borided layer. The artificial neural network is shown schematically in Figure. 2.
The activation function used is the sigmoid function given by the following expression:
The learning algorithm is a mathematical method that changes the connection weights to converge to a solution that allows the network to perform the desired task. The learning is a parametric identification, which optimizes the weight values of the network 20-21. During this phase, the behaviour of the network is changed until getting the desired one.
In this study, we use the classical supervised back propagation learning method, which is a learning algorithm suitable for multi-layer neural networks.
Back propagation is the best known learning method, and one of the most efficient for multilayer networks 20.
Learning algorithms are often iterative; they adapt the connection weights after the presentation of each input vector. It is necessary to set the input data many times until the weights converge to stable values.
A training base is the test base, which performs the training of the network and it is used to find a set of optimized weights. The network was trained with the experimental data obtained from the boriding of AISI 316L steel 22.
The weights values (wik) and values are automatically initialized with a program that we wrote in the C++ language. The number of iterations performed during the training phase is 60000 iterations, during the training phase and the learning rate was set to 0.04.
3. Experimental procedure
In order to test the validity of mathematical model, we used the results of the boriding experiments on AISI 316L steel taken from our own experimental data recently published in 22. In that experiment, the samples of AISI 316L stainless steel were selected for boriding treatment.
The chemical composition of AISI 316 L steel ( in mass % ) is the following: : 0.03% C, 1.3% Mn, 12.2% Ni, 0.35% Co,17.4% Cr,2.28% Mo,0.44% Ti, 0.45% Si and 0.07% V.
The boriding process were achieved in molten salts, constituted of sodium tetraborate Na2B4O7 (70% in mass) and a reducing agent silicon carbide SiC (30% in mass).
The use of the silicon carbide (SiC) as a reducing agent led to a single-phase layer (Fe2B). The thermochemical treatment was done at the three different temperatures 850°C, 950°C and 1000°C with three treatment times 2, 4,6 and 8 hours.
After the boriding treatments, some samples were sectioned longitudinally to obtain two sections for optical microscopy observation. It has been shown that the properties of the diffusion layers depend on their thickness, chemical structure, phase composition and kinetic parameters used in the boriding process.

Results and discussion
In the first section, we presented the relative results for the first model of artificial neural network, which we compared to the experimental results 22. In the second section, we presented the simulation with mathematical model based on Fick’s second law 13 and the comparison of the results obtained by these two models.
Artificial neural network approach
The learning of artificial network was made via the same experimental results 22 used in the first model with 60000 iterations. The convergence rate value of the network is 10-6. To apply the learning algorithm of the network we normalized the experimental data with the following parabolic equation:
With : the thickness of the Fe2B borided layer, : the kinetic constant and t: boriding time.
The learning algorithm purpose is to provide a method to the network, so it can adjust its settings for examples’ treatment.
Learning is the process of adapting the parameters of a system to give a desired response to any input or to any stimulation. In Table.1, we gathered the values of borided layer thicknesses obtained by the method of artificial neural networks and their comparison with the experimental results.
We found a good agreement between the experimental data and the data obtained with the neural network at different temperatures.
The mean prediction error of the thickness of borided layer related to different treatments from 850 °C to 1000 °C as a function of time is plotted on figure.3. We observed that the error shifts between 1 and 1.5 µm for different processing times.

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Mathematical model
The mathematical model used is derived from a previous work by Mebarek et al. 13. This model is based on the solution of Fick’s diffusion equation in a semi-infinite medium on one hand, and on the assumption that the boriding process is an equilibrium process on the other hand. The local thermodynamic equilibrium is quickly reached at each point in the material, from which we can estimate the growth rate at the interface (Fe2B/?-Fe) and determine the boride layer thickness. The important parameters for the simulation are the temperature, process time, the diffusivity of boron in each phase and the concentration of boron at the material surface.
For the Fe2B phase, we calculated the diffusion coefficient with the method given by Bektes et al. 23. The relation between the boride layer thickness and the boriding temperature is given by:
where, d is the experimental boride layer thickness (?m), D0 is the boron diffusion coefficient (?m2/s), t is the boriding time (s), Q is the -activation energy for boron diffusion (J/mol), R is the universal gas constant R=8.314 J/mol K, and T is the boriding temperature (K).
Equation 4 can be written as follows:

x

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