Forecasting the Productivity of the Agrophytocenoses of the Miscanthus Giganteus for the Fertilization Based on the Wastewater Sedimentation Using Artificial Neural Networks

The observations of plant development were carried out for three years. The most desirable period for harvesting the miscanthus is December. During this period, the humidity of the stems decreases to 17%. For this reason, the samples for laboratory tests were taken in December. According to the obtained research data, the sewage sludge used is characterized by the following indicators: humidity – 76%, ash content – 5%, nitrogen – 0.66%, P2O5 – 2.51%, K2O v 2.16%. In this study, a mathematical model which allowed predicting the yield of the miscanthus at given levels, with the introduction of the mineral and organic (sewage sludge) fertilizers was successfully implemented. According to the performed research, the application of a sewage sludge in norm of 20–40 t/ha promotes the productivity of the power cultures (the miscanthus) within 24.5–27.1 t/ha, thus increasing productivity on 2.3–5.1 t/ha, compared with control.


INTRODUCTION
The energy needs of the modern world are too great, and their scale has approached a state where not only economic but also natural factors limit the involvement of the traditional energy resources. Already at the beginning of the 21 st century, up to 1 billion tons of the conventional fuel of the vegetable mass have been used for the energy purposes, which is equivalent to 25% of the world oil production (Vambol 2018, Tryboi 2018, Rojas 2013, Haykin 2009). The energy crops are an effective way to solve the problems with the longterm supply of the raw materials. According to the estimates of the Bioenergy Association of Ukraine, the potential of the bioenergy is 21 million tons of the energy, including about a 7 million tons of the energy crops on 2 million hectares of the land (Vambol 2018, Specht 1990, Zheliezna 2014).
In the developed countries of the world (USA, China, Germany, Great Britain, Denmark, Australia, Spain, Poland, etc.) the production and use of the energy crops as biofuels is actively developing. Various crops (corn, rapeseed, winter wheat, sunflower, willow, poplar, etc.) are used as sources of the organic biomass. One of the nontraditional energy crops that has been of interest to researchers in the last two decades is the miscanthus (Nussbaumer 2004 The miscanthus, which is huge in terms of the dry biomass yield, solar energy efficiency and environmentally friendly cultivation technology, has certain advantages over other energy crops. In the available professional and popular science literature, the miscanthus is positioned as a crop capable of providing high yields of the biomass -about 20-30 tons of the dry matter per hectare (Fantozzi 2010, Alexopoulou 2012, Cherubini 2011, Heller 2003. It is not particularly demanding in terms of soil fertility, so it is suitable for growing in poor lands, where traditional crops cannot bring good yields. It is similar to appearance to corn or reeds. This is a perennial plant, which, having seeds, reproduces exclusively by parts of the rhizomes -rice (Alexopoulou 2010, Cherubini and Strømman 2011, Heller 2013, Fantozzi 2010, Lopushniak 2020a).
The degraded soils, due to their characteristics, are optimal for growing energy plantations. They are well supplied with moisture and nitrogen, allowing plants to accumulate a fairly substantial biomass with moderate application of the mineral fertilizers. In addition, the cultivation of the continuous crops on contaminated, degraded soils is an important factor in the ecologically balanced use of these lands (Lopushniak 2018b, Karlik 2011, Felten 2012.
The purpose of the research is to develop and evaluate a methodology for predicting the productivity of agrophytocenoses of the giant miscanthus with the introduction of the sewage sludge using artificial neural networks.

MATERIALS AND METHODS
The research was carried out in the Tysmenytsia district of the Ivano-Frankivsk region on sod-podzolic degraded soils on the territory of the Maidan settlement (Tsenzhiv station). The studies of the chemical composition of the experimental samples were performed by X-ray fluorescence analysis. The method is based on the analysis of the fluorescence spectra of the elements emitted during the adsorption of the high-energy radiation. The method allows obtaining the data on the chemical composition of the substance in a wide range with an accuracy of 1-10 ppm. The experiments were performed on a precision analyzer EXPERT 3L with a constant supply of the helium into the channels of the collimator. These studies allowed controlling the agrochemical parameters of the soils with an accuracy of 0.005% (Calderon 2017, Cherubini 2011).
Options 3-8 are balanced by the introduction of the main elements of the mineral nutrition -NPK (Lopushniak 2020).
The miscanthus was planted by hand according to the scheme 50 x 70 cm. The width of the experimental plot is 5.0 m; length -7.0 m; accounting area -35.0 m 2 . The total area of the experimental plot is 63 m 2 . The placement of the plots in three repetitions is systematic. All the work, except for soil preparation, was carried out manually. The rhizomes with 5 -6 growth buds were used for better rooting. The depth of the rhizome making is from 12 cm (Dagl, 2012).
Usually, in the first years, the plant forms a strong root system. The lines do not close. In the first years of the growing season at the beginning of the regrowth at plant heights of 15-20 cm loosening between rows was carried out. The plant is resistant to frost, due to a significant increase in rhizome and the number of the aboveground shoots, the continuous herbage is formed. During this period, the miscanthus is very competitive with weeds and does not require row spacing (Felten 2012., Grytsulyak 2016, Dondini 2009).
The observations of plant development were carried out for three years. The most desirable period for harvesting the miscanthus is December. During this period, the humidity of the stems decreases to 17%. For this reason, the samples for laboratory tests were taken in December (Rojas 2013). According to the obtained research data, the sewage sludge we used is characterized by the following indicators: humidity -76%, ash content -5%, nitrogen -0.66%, P 2 O 5 -2.51%, K 2 O -2.16%.
Various approaches and methods are currently used to predict crop yields. These are mainly methods based on statistical processing of the data obtained in previous years and the subsequent approximation of these data to a certain mathematical model (Haykin 2009). Modern information technology makes it possible to solve this problem using elements of the artificial search, namely artificial neural networks. An artificial neural network (ANN) is a mathematical model that is similar in structure to the human nervous system. Like humans, ANN can learn and generalize knowledge. That is why ANN is referred to as artificial intelligence. ANN is widely used in all fields of the science. There are many different types of ANNs that focus on the specific tasks (Rojas 2013).
In general, ANN is a system of the interconnected and interacting processors (neurons). A neuron (the basic element of ANN) is a simple computing processor that can receive, process, and transmit information. When a large number of the neurons are combined into a single network, the system can solve non-trivial problems.
The neurons in ANN are combined into layers (Fig. 1) -the input layer (the set of neurons that receive information), n hidden layers (the set of the neurons that process information), and the output layer (neurons which output the result).
During the operation, the neurons operate on numbers. Usually, these numbers are in the range [0, 1] or [-1, 1]. Each neuron has two parameters: input data and output data. The input data field contains the summary information from all the neurons in the previous layer. After receiving the data, the information is normalized using the activation function f (x), after which it enters the output data field. It should be noted that for the input layer of the neurons, the input information is equal to the output (input = output).
All the neurons are connected to a network by synapses (Fig. 2, Wi, Wj). A synapse is a connection between two neurons that is characterized by the weight of the synapse [14].
Due to the weight of the synapse, the input information changes during the transmission from one neuron to another. Figure 2 shows how the weight of the synapse can change the data on the example of the colors. During the processing of the input data, a result will be obtained in which the most important role will be played by the synapse with the greatest weight. The totality of all weights of the neuronal synapses allows the system to make decisions. Depending on the complexity of the task, the number of the neurons and layers can vary greatly.
Another important element of the neural network is the activation function ( [15]) -a function that normalizes the input data (i.e. a function that allows interpreting the data in the form of the numbers belonging to the range [0, 1]). In the activation function to determine the output data, the total amount of the input data and weights  is compared with some threshold. If the sum is greater than the threshold value, the processing element generates a signal. Otherwise, the signal is not generated (or a brake signal is generated) (Karlik and Olgac 2011). The sigmoid function is the most often used in practice (Fig. 3). An important feature of the sigmoid is the continuity of the functions and their derivatives. In the case of the use of non-numerical data, they are structured on categories ( Table 1).
The power of the modern computers allows handling quite complex tasks, even on non-specialized devices. The interested parties can create and teach ANNs on regular computers using the publicly available software. There are a large number of the software packages for working with ANN (Dagli, C. H. (Ed.). 2012). Most of them are aimed at an experienced user who has programming skills.
Among the software with a low entry threshold, one can be including STATISTICA (Karlik and Olgac 2011). This program allows quickly building ANN of different architecture and complexity. It is also important that after learning ANN in this program, one can immediately conduct research on new data and obtain results instantly.

RESULTS AND DISCUSSION
The STATISTICA program was used to build and train artificial neural networks. The input data were the norms of the mineral fertilizers, sewage sludge and compost based on them, as well as the obtained result of the raw mass of the miscanthus. Table 2 shows a diagram of the experiment for one of the groups of the sites. The data presented in the table represent the amount of the application of a particular type of the fertilizer and the corresponding yield. Thus, N, P 2 O 5 , K 2 O -applied with the mineral fertilizers N + SS, P 2 O 5 + SS, K 2 O + SS -applied with the organic fertilizers (sewage sludge + straw), Mass -the mass of the miscanthus (t/ha). Under Manure, respectively, it is meant a set of the fertilizers of the studied options. From each variant, 5 independent samples were taken accordingly for more exact mathematical processing. Accordingly, we obtained 40 results ( Table 2). The total amount of the fertilizers was: For 1 -without fertilizers, 2 -180, all others -280 -the sum of the basic agrochemical indicators. A general view of the experimental data used for ANN is shown in Figure 4. Table 3 shows the basic statistical evaluation of the data that served as input for the training of ANN. This study is performed to estimate, if necessary, the average value of the variables and their standard deviation. The training of the neural networks was carried out using the strategy of building the ANS model (automated neural network) ( Table 3).
The training of the neural networks was conducted using the strategy of building the ANS model (automated neural network). By the strategy of the selection of the subsamples, 5 random subsamples were chosen with a relative percentage: educational (70%) -control (15%) -test (15%). Logistic, Tanh (logistic and hyperbolic functions) were selected as activation functions  Note: The number in order corresponds to a certain rate of the fertilizer. The Data ji -data recorded in a particular case and grouped by the appropriate options.   for the source neurons. The selected activation functions for hidden neurons were Logistic, Tanh and Exponential (logistic, hyperbolic, and exponential). Moreover, the number of the hidden neurons was selected as input parameters -from 3 to 20. The decaying regularization of the weight of the neurons (weight decay) -from 0.001 to 0.01 (for hidden layers) was set. The number of networks trained is 20. These parameters were selected experimentally. In general, the parameters are selected depending on the complexity of the problem and can vary widely (http://neuralnetworksanddeeplearning.com/). After the learning ANN, 5 ANN models were obtained, which had different performance indicators. The productivity in this case is the percentage of the correct prediction in the test sample (from 0 to 100%). Among the obtained ANNs, a network with optimal performance was selected. Figure 5 shows the ANN obtained after training and the network with optimal performance (network No. 5). MLP 6-3-1 (6 input, 3 hidden, 1 output neurons).
It is worth noting that the forecast results are often not satisfactory; then, one can improve the result by going to the CNN model, where the refinement input can be set (namely -RBF network type -radial basis function, number of the training networks, number of the hidden neurons and number of the epochs). Such operations can be performed until the test performance and learning performance will suit us, according to the performance data.
The sensitivity analysis of the variables included in the model is also important. In table 4, it is shown the results of the sensitivity assessment.
According to the obtained results, the largest contribution to the model is made by the variable P 2 O 5 + SS (option 8). In view of this, in further research and change of the settings, it is also necessary to pay special attention to this option (Compost (SS + straw (3: 1)) -30 t/ha + N 30 K 55 ) fertilization of the energy crops.
The result was ANN, which gives good prediction results on a test sample. The software package allows checking other data, i.e. entering other options for the ratio of the fertilizers and obtaining a forecast by weight. For example, the data on the fertilizers were introduced -N-8, P 2 О 5 -12, K 2 О -50, N + S -82, P 2 О 5 + SS-68, K 2 О + SS-32. The yield of the forecast model is 25 t/ha, which is in good agreement with the experimental data. In Figure 6 is shown the result of the processing this example. Figure 7 shows the relationship between the experimental data (target axis) and the expected data (output axis) on the test and training sample. As it can be seen from the figure, the results predicted by ANN are not close to the experimental ones, which confirm the quality of the model. The blue dots in Figure 7 show the results of the study of the miscanthus productivity, for the introduction of the sewage sludge at a rate of 20-40 t/ha and compost based on them in the range (22.4-27.1 t/ha). Basically, the forecast data fall on the

CONCLUSIONS.
The artificial neural networks are widely used in various fields of knowledge, namely to predict the productivity of the agrophytocenoses of the energy crops. This research technology -artificial neural networks -is a mathematical model that allows finding the relationships between variables and predicted results of the studied variables, depending on the initial conditions. According to the obtained results, it can be successfully used to predict the productivity of the different energy crops in line with the given input parameters (fertilizer rates, basic agrochemical indicators, crop productivity). In this study, a mathematical model which allowed predicting the yield of the miscanthus at given levels, with the introduction of the mineral and organic (sewage sludge) fertilizers, was successfully implemented. According to the  performed studies, the application of a sewage sludge in norm of 20-40 t/ha promotes productivity of the power cultures (the miscanthus) within 24.5-27.1 t/ha; thus, it increases productivity on 2.3-5.1 t/ha, compared with control.