Simulation model parameter optimization method for multidimensional signals

The paper describes an approach to minimize the number of simulation experiments of multidimensional signals by means of a regression neural network. Multivariate signal simulation systems are in demand for testing real-time computing systems, but they mostly have a wide vector of model parameters. The formation of parameter vector for the simulation model of multidimensional signals, which ensures an adequate solution to the problem of signal processing obtained as a result of the model operation, is an urgent task. The authors propose a method of heuristic optimization of input parameters, implemented by means of machine learning, which reduces the searching of values for the given optimization function. In this study the ability to perform real-time signal simulations is the optimization feature. The research outcomes are the description of the neural network, the rationale for its configuration, and the training and validation results.


Introduction
Digital signal processing is an integral part of systems for collecting and processing information on various physical processes.Such systems are a sort of intermediary in the communication between human and the natural environment, allowing him to interpret the readings of various sensors (microphone, video camera, antenna arrays) quickly and accurately, using modern computational tools.Processing of sound, images, video fragments; radar, hydroacoustic and ultrasonic signals has become the basis for building control systems for various complex technical objects in the field of entertainment, security and medicine [1,2].The quality of any signal processing system directly depends on the elaboration of its testing technology.The basis of any signal processing system testing technology, in its turn, is a pre-prepared sufficiently large set of input test data describing physical processes perceived by the system sensors.
Input signals have to cover many different variations of external conditions and factors, including different noise environments.The technological data should optimally be obtained as a result of recording the results of physical experiments with various parameters of the object under study and external conditions, and the minimum required number of such sets of input signals should be at least one million according to expert estimates.
In many practical fields of science and technology, obtaining such a quantity of data in the process of in-situ tests is a practically unattainable task due not only to the huge labor intensity and high cost of experiments, but also due to physical and technical limitations of the ability to control the parameters of the physical environment of signal propagation, weather conditions and other external conditions of the experiment.
An effective method of solving the limitations of experiment parameters control and essential acceleration of the obtaining the necessary amount of input data process is the simulation of signals [3].In this case, a necessary requirement for the signal synthesis system along with the adequacy of the model is the performance of data generation, since a critical requirement in the validation of digital signal processing system is its operation in real time.In those cases, when the increase in performance by increasing the computing power of hardware reaches its limit due to cost or technical limitations, developers can turn to the method of increasing the performance of the system due to a justified decrease in the accuracy of the synthesized signals [4,5] by manually adjusting the parameters of the simulation model.Obviously, the reduction of the model accuracy should not lead to a loss of its adequacy.
In a real complex technical system, the number of such parameters can be quite large, and the effect of increasing system performance from reducing one particular parameter has no analytical description, so manual adjustment of simulation model parameters requires a high level of expert knowledge from the model developer, a lot of time and in general cannot guarantee the optimal result in a limited amount of time.
In recent decades, there has been a significant expansion of the range of scientific and technical problems successfully solved with the help of neural network technologies.Such a breakthrough has become possible due to the recently significantly increased computing capability of data processing hardware and simultaneous development of algorithmic and software tools of neural networks.The greatest effect from the application of neural network technologies was achieved in those areas where mathematical and algorithmic support of the task solution is difficult to be formalized.
In this paper the authors propose a method of adaptive control of parameters of the multidimensional signal simulation model, implemented by means of machine learning, which allows to increase the performance of signal generation, thus ensuring its real-time operation at the expense of adequate reduction of accuracy.
The results of the conducted research are the description of the neural network, the rationale for its configuration, and the training and validation results.

Neural network structure specification
Artificial neural network (ANN) is a mathematical model, as well as its software or hardware implementation, built on the principle of organization and functioning of biological neural networks -networks of nerve cells of living cells.
ANNs can be categorized into two types: feedforward and feedback networks.The first one has the advantage of small net: easy learning process and simultaneously a neurons are repeatedly involved in data processing, lack of context memory as the disadvantage.The feedbacks in structure allow us to have learning complexity caused by the large number of neurons for algorithms of the same complexity level but need special conditions guaranteeing convergence of calculations.
Recurrent neural networks (RNN) are a subspecies of feedback networks.A characteristic feature of such networks is the presence of dynamic delay and feedback blocks, which allows them to process dynamic models.It contains a wide memory window (context vector) in which old values are gradually flushed.
Long short-term memory (LSTM) is a special kind of recurrent neural network architecture capable of learning long-term dependencies.These neural networks allow solving many different tasks, such as text analysis, automatic translation, automatic speech recognition and others [6].
LSTM is suitable for time series prediction because it is analogous to a short-time window, allowing the existing context to be preserved while predicting the next values.
The LSTM operation consists of four steps (figure 1): • Information exclusion (forget gate): • Preserving the information (cell state): • Updating the information (input gate): • Formation of output information (output gate): Connecting the outputs of the cell to the inputs we get a network of cells (figure 2).
where p i is the number of parameter i possible values, N -modeling parameters number as it is shown in figure 3. The neural network is trained simultaneously with the new values being chosen in the direction of decreasing the error gradient.Each step involves finding a new direction for changing the model parameters with values that are the input for the LSTM network.
The output of the neural network is a time-domain multidimensional signal, which is analyzed for its adequacy according to a given criterion (optimization function).In a particular case, this function is the ability to conduct modeling in real time.
During the learning on each step we get closer to optimized value of chosen optimization function.In particular case, it is realtime (RT) and non-realtime (NRT) functioning of simulation model.
The algorithm 1 describes the core learning process for the posed problem.• Φ M -operator of signal transformation when passing the physical wave propagation medium (attenuation, noisiness) • Φ 0 -operator of signal transformation at reflection of probing signal from heterogeneities (communication object, boundaries of physical media of wave propagation, etc.), • P M -vector of parameters of the physical medium of wave propagation, • P 0 -vector of parameters of heterogeneities of the physical medium of wave propagation, • Φ R -operator of signal transformation inside the receiver equipment (natural noise of the receiving channel, quadratic processing, etc.).
In previous studies [7,8], this structural scheme was achieved by optimizing a generalized simulation scheme for digital communication process simulation because the magnitude of the initial phase and the nonlinear distortions of active interference in the propagation of its waves in the physical medium do not affect the adequacy of the entire model.
Nevertheless, even considering the optimized simulation model, there is no analytical correlation between the computational complexity of the model (real-time execution capability) and the parameter vectors P L , P DC , P N S , P N i , P M .In this case, it became necessary to use machine learning to improve the quality of the simulation.
The error and loss functions training result is shown in the figure 5.
where N -number of counts, X(t) -actual value of time series, X(t) -predicted value of time series.Mean absolute percentage error (MAPE) shows the magnitude of the error relative to the values of the series.It is useful for comparison of performance of a predictive model on different series or performance of different models on one series: Weighted absolute percentage error (WAPE) is a statistical metric for evaluating forecast accuracy, typically for time series data.It is commonly used for measuring a model's performance when the dataset used has low or intermittent values: To find the best practices of parameter optimization the number of epoch during training was adjusted.The results of calculated metrics can be found in table 1.As can be seen from the table, as the number of epochs increases, the errors decrease, but after a certain value there is a so-called overtraining, as a consequence of which there is an accidental decrease in accuracy.
In the figure 6 you can see the comparison of the predicted X(t) and calculated X(t) results.
The number of epochs equal to 250 was chosen in the problems of parameter adaptation of multidimensional signal modeling based on the PXI system from National Instruments, since the accuracy of such a model is adequate and the search time for optimized parameters is significantly reduced.

Conclusion
As a result of the research carried out, the authors present a method for optimizing the parmameters of the simulation model.
The results of the study showed the effectiveness of this method in the generation of multidimensional signals, under conditions of a large number of model parameters.The The developed method can be used within the framework of testing control systems of various complex technical objects.The experimental processing of simulated signals has been carried out with the onboard control system of information processing complex designed in Bauman Moscow State Technical University.
As perspectives for this study, we can consider using modified generative GAN models for this task without the classical discriminator that characterizes teacherless learning.Also, within the framework of control system testing tasks, it would be useful to cluster the obtained optimized simulation parameters to create intervals and determine the limits of performance of the tested object.

Figure 4 .5
Figure 4. Structural diagram of the generalized digital communication simulation model

Figure 6 .
Figure 6.Prediction compared to real data graph

Table 1 .
Accuracy metrics of different training epoch number