# Comparing Polynomials and Neural Network to Modelling Injection Dosages in Modern CI Engines
## 4 Discussion  
The presented method of injection dose prediction can be used to support the diagnostic processes of modern fuel injectors, especially piezoelectric ones. The element controlling the valve operation of such a fuel injector is a pile of crystals.  
The stack of crystals (Figure 17) consists of 350 piezoelectric elements. After applying the voltage, each of them extends by 13 \(\upmu\)m, in total the entire element changes its length by 4550 \(\upmu\)m. The test of the stack consists in measuring its resistance at a voltage of 150 V. If the resistance is 180-200 k\(\Omega\), the stack is functional. Unfortunately, sometimes despite the tests carried out, where the result was normal and the fuel injector did not work properly due to the ailments of individual crystals.  
The research showed that the values of the deviation of individual crystals change during operation. This leads to an anomaly during the operation of the fuel injectors within a certain range. Detecting this type of fault is laborious, therefore the use of the numerical method may speed up the process.  
In the article, the authors used two methods to determine the characteristics of the injector on the basis of measurement data from the test bench. The first one is polynomial interpolation, where a polynomial has been determined using an appropriate algorithm, which determines the performance characteristics of the tested object. Analyzing the results of calculations, the mean error is 0.56976 mm\({}^{3}\)/H, the maximum error is 4.34 mm\({}^{3}\)/H and the minimum one is 0.77 mm\({}^{3}\)/H (Table 3).  
Figure 16: Injection dose size characteristics obtained as a result of the implementation of a neural network.  
Figure 17: Piezoelectric crystals stack.  
By analyzing the results of the calculations, it can be concluded that the average injection dose errors determined by the polynomial and the neural network are approximate. A minimally smaller deviation occurs when the neural network is used. It is worth noting that both in the neural network and in the polynomial, the greatest errors appear at the highest loads of the injector. The solution may be to extend the characteristics of the injector in order to provide a larger database necessary for learning the neural network and determining the polynomial. The use of the polynomial approximation method is a better solution having only two input parameters and one output parameter. The prediction does not depend on the randomness of the initial data, such as weights in the neural network. Every time an attempt is made to generate a polynomial, the same result is always obtained. However, in the case of diagnostics, where there are more than two input parameters, a much better method is to use a neural network. Nevertheless, this method requires more analysis in terms of the selection of the number of hidden layers, the number of neurons and the selection of the method of backward error propagation. It is often necessary to analyze multiple combinations of structures in order to assess which combination is the most optimal and most likely to be successful for diagnostic purposes and to correctly predict the performance of the injector under given conditions. In addition, a serious disadvantage of this method is the large influence of randomly selected weights at the initial stage of learning the neural network. The application of computational methods, in particular neural networks, is an innovative method that has not been used yet in the field of diagnostics and prediction of fuel injectors operation.