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This paper presents an automatic system of neural networks (NNs) that has the ability to simulate and predict many of applied problems. The system architectures are automatically reorganized and the experimental process starts again, if the required performance is not reached. This processing is continued until the performance obtained. This system is first applied and tested on the two spiral problem; it shows that excellent generalization performance obtained by classifying all points of the two-spirals correctly. After that, it is applied and tested on the shear stress and the pressure drop problem across the short orifice die as a function of shear rate at different mean pressures for linear low-density polyethylene copolymer (LLDPE) at

Neural networks are widely used for solving many problems in most science problems of linear and nonlinear cases [

Neural classifiers can deal with many multivariable nonlinear problems for which an accurate analytical solution is difficult to obtain.It is found however that the use of neural classifiers depends on several parameters that are crucial to the accurate predictions of the properties sought. The appropriate neural architecture, the number of hidden layers, and the number of neurons in each hidden layer are issues that can greatly affect the accuracy of the prediction. Unfortunately, there is no direct method to specify these factors as they need to be determined on experimental and trial basis [

The two-spiral benchmark was considered as one of the most difficult problems in two-class pattern classification field due to the complicated decision boundary [

The effects of pressure on the viscosity and flow stability of one of the commercial grade polyethylenes (PEs) which is linear low-density polyethylene copolymer have been studied. The range of shear rates considered covers both stable and unstable flow regimes. “Enhanced exit-pressure” experiments have been performed attaining pressures of the order of

Very high pressures can be exerted on polymers during processing. At these pressure levels, polymer melt properties, and flow stability, evolve according to laws that are different from those used at moderate pressures. In the following work by Couch and Binding [

BP is the most widely used algorithm for supervised learning with multilayered feed-forward networks [

The RPROP algorithm was faster than the BP [

The two-spiral problem is a classification task that consists of deciding in which of two interlocking spiral-shaped regions a given coordinate lies. The interlocking spiral shapes are chosen for this problem because they are not linearly separable. Finding a neural network solution to the two-spirals problem has proven to be very difficult when using a traditional gradient-descent learning method such as backpropagation, and therefore it has been used in a number of studies to test new learning methods; see for instance, [

To learn and solve this task, a training set consists of 194 preclassified coordinates. Half of the coordinates is located in one spiral-shaped region and marked with triangles, and the other spiral-shaped region marked with circles. The coordinates of the 97 triangles are generated using the following equations, where

When performing a correct classification, the neural network takes two inputs corresponding to an

The studied problem consists of two dependent parts, the first is the pressure drop across the short orifice die as a function of shear rate at different mean pressures for linear low-density polyethylene copolymer at

This problem has two inputs (mean pressure and

A block diagram modeling.

Neural networks consist of a number of units (neurons) which are connected by weighted links. These units are typically organised in several layers, namely, an input layer, one or more hidden layers, and an output layer. The input layer receives an external activation vector and passes it via weighted connections to the units in the first hidden layer. Figure

Network Architecture for one hidden layer.

In the RPROP algorithm, each weight (

The size of the weight change is exclusively determined by the weight-specific update-value

As mentioned in the end of Section

The proposed system is designed to work in automatic way starting with random initial weighed and biases values. Many NN experiments are done to have the optimal NN results of the two-spiral problem, by repeating the same experiment using the same NN architecture (number of hidden layers and neurons). Therefore, 500 NN experiments and 400 neurons are specified as maximum numbers of this system. The system stops when the best network is obtained. This system is trained and tested using different parameters, for instance, changing the number of hidden layers, neurons, and epochs. The experimental data of the two physical problems (the shear stress and the pressure drop problem across the short orifice die as a function of shear rate at different mean pressures for linear low-density polyethylene copolymer (LLDPE) at

The proposed system diagram.

This proposed system is based on RPROP algorithm using tan-sigmoid transfer function in the hidden layers and a linear transfer function in the output layer. More hidden layers or neurons require more computations, but allow the network to solve complicated problems. Therefore, many tries are done to find the best network that uses low number of hidden layers and low number of neurons.

After the training, in the test process of the two-spirals problem, it is noticed that the chosen algorithm using two hidden layers with 77 neurons for each one is very effective for reaching the optimal classification; see Figure

The architecture of the proposed network.

Two-Spirals problem.

linear low-density polyethylene copolymer problem

The proposed system is carried out on three problems. They are two-spiral problem, the pressure drop and shear stress across the short orifice die as a function of shear rate at different mean pressures for linear low-density polyethylene copolymer at

This problem is used to learn a mapping function (two inputs and one output) which distinguishes points on two intertwined spirals. This is one of the typical difficult problems due to its extreme nonlinearity. The proposed system was first trained on 194 points of the

The NN evaluations.

Experiments | ▴ | • | Accuracy |
---|---|---|---|

1-HL & 400 neurons | 6 | 8 | 92.8% |

2-HL & 50 neurons | 6 | 2 | 95.9% |

2-HL & 60 neurons | 1 | 0 | 99.5% |

2-HL & 77 neurons | 0 | 0 | 100% |

▴ means no. misclassified of the triangles spiral.

• means no. misclassified of the circles spiral.

The produced weight and bias values of the best trained network.

Pressure drop | Shear sftress | ||||||||

Weights | Biases | Weights | Biases | ||||||

IW | LW | b | b | IW | LW | b | b | ||

–0.0092357 | –0.074211 | –0.91753 | 16.5516 | 1.0932 | 0.42093 | 0.11045 | –0.29861 | –14.8788 | 0.49395 |

–0.064121 | –0.075615 | –1.238 | –14.7538 | 0.14907 | –0.028838 | 0.14388 | –12.3761 | ||

0.0086149 | –0.025249 | –2.0174 | –2.5568 | 0.00031069 | 0.026445 | –0.52767 | –11.8835 | ||

–0.0017662 | 0.036417 | 1.4997 | –2.6596 | –0.0013544 | 0.0019265 | –0.71031 | –1.7975 | ||

0.023687 | –0.0099672 | 0.58772 | –3.0918 | –2.385 | –2.4141 | 0.38584 | –7.6139 | ||

–0.017128 | 0.0023562 | –0.97796 | 7.6266 | 0.01941 | 0.0086871 | 0.089209 | –11.4316 | ||

–0.063098 | –0.083557 | –1.6479 | –8.8354 | 0.0035304 | –0.075417 | –0.67559 | –1.0939 | ||

0.20564 | 0.021342 | 0.91681 | –6.5694 | –0.0076158 | –0.028539 | –0.19949 | 1.5115 | ||

–0.003303 | 0.022274 | 1.7385 | 0.081723 | 0.044023 | 0.038829 | 0.017871 | –10.0832 | ||

–0.041185 | 0.079807 | 1.8449 | –11.2574 | 0.005363 | –0.024715 | –0.49156 | 6.6207 | ||

0.024424 | –0.075788 | –1.1288 | –2.3592 | –0.0013665 | 0.032209 | 0.18869 | 7.0804 | ||

0.17343 | 0.14075 | 1.9772 | 1.7446 | –0.00048342 | –0.00098168 | –2.8782 | 0.41403 | ||

–0.011373 | –0.02235 | –2.4026 | 9.3959 | 0.00076532 | –0.040128 | –0.65477 | –1.5022 | ||

0.0038815 | 0.082769 | 0.97819 | 8.3238 | –0.0081859 | 0.27396 | 0.15231 | –4.8667 | ||

0.0045642 | 0.081391 | 0.95285 | 7.9681 | –0.012168 | –0.012549 | –0.14515 | 6.2695 | ||

–0.0023447 | –0.011416 | –3.3245 | 6.9351 | 0.0038695 | 0.010562 | –0.29916 | –3.3411 | ||

–0.17721 | –0.15542 | –1.3838 | 1.5062 | –0.054214 | –0.0096744 | –0.19168 | 4.0609 | ||

0.013042 | 0.020739 | 0.42706 | –13.5254 | –0.014118 | 0.015177 | 0.30145 | –5.2238 | ||

0.094386 | 0.060478 | 1.6078 | 11.2445 | –0.0014922 | 0.024903 | 0.32635 | –1.9256 | ||

0.10412 | 0.10144 | 0.4335 | 5.2448 | 0.0046948 | –0.030181 | –0.32884 | –0.096699 |

LW

The training process is continued with increasing one hidden layer more. The obtained performances are 95.9%, 99.5%, and 100% using 50, 60, and 77 neurons, respectively; see Figures

The NN performance (HL means hidden layer).

1-HL & 400 neurons

2-HL & 50 neurons

2-HL & 60 neurons

2-HL & 77 neurons

The NN simulation of the pressure drop dependence on shear rate.

1-HL & 400 neurons

2-HL & 50 neurons

2-HL & 60 neurons

2-HL & 77 neurons

The above mentioned details of the proposed system were applied and simulated to the data of the pressure drop and shear stress across the short orifice die as a function of shear rate at different mean pressures for linear low-density polyethylene copolymer at

The system was trained using the chosen neural network on six cases of different mean pressures for each of the pressure drop and shear stress as a function of shear rate. These values of mean pressure are 1, 100, 200, 300, 500, and 600 multiplied by 10^{5} Pa. The performances of the obtained networks are shown in Figure

The NN performance

The pressure drop problem

The shear stress problemf

The NN simulation of the pressure drop dependence on shear rate.

The NN simulation of the share stress dependence on shear rate.

Figure

Figure

The proposed system is automatically designed to find the best network that has the ability to have the best test and prediction. This technique is started by doing 500 NN experiments with incrementing the number of neurons for each hidden layer. In the incrementing process, another new 500 NN-experiments are carried out in alternative way; the number of hidden layers is incremented by one with initializing the number of neurons for these hidden layers and new 500 NN-experiments are done. This process is continued until the required performance is reached. Therefore, many tries are automatically done to find this network, using low number of hidden layers and neurons.

The obtained performance of the two-spiral problem is low when using one hidden layer in the network architecture, although the number of neurons increased up to 400. The performance is improved using two hidden layers, it is 95.9% with 50 neurons, 99.5% with 60 neurons, and 100% with 77 neurons. In the best performance, all points of the two-spiral problem are correctly classified.

In the other two problems, it was found that one hidden layer with 20 neurons is enough for reaching the optimal solution. The trained NN using this system shows excellent results matched with the experimental data in the two cases of shear stress and pressure drop problems. The NN technique has been also designed to simulate the other distributions not presented in the training set and matched them effectively.

The NNs simulation using RPROP algorithm is powerful mechanism for classifying all points of the two spirals, and for the prediction flow curves (dependence of shear stress on shear rate) and pressure drop dependence of shear rate at a certain value of mean pressure across short orifice die for linear low-density polyethylene copolymer at