**6. Conclusions and Perspectives**

The paper provides mathematical formulations of the machine learning problem, supervised and unsupervised, defines the basic concepts, such as the work area and the training set. Based on the presented formulations, it is shown that the main task of machine learning is to find a function that determines the correspondence between the input data and the resulting data. It is shown that today this problem can be solved numerically using symbolic regression methods. The problem of obtaining a mathematical expression arises in various situations—approximation of experimental data to determine a physical law or a trend model; efficiently analyze and predict variables or indicators based on previous observations; identification of a mathematical model of a process or a dynamic object; generalization of the control law based on the current state of the control object. The application of machine learning based on symbolic regression methods to control opens up the possibility of solving such a complex problem in control theory as the problem of general control synthesis. The paper presents a mathematical formulation of the control synthesis problem and provides methods for its solution using machine learning both directly and based on a training set. An important result of the article is the methodology for solving the problem of general control synthesis as machine learning control based on a training set. An approach to constructing a training sample based on multiple solutions to the optimal control problem is proposed. An example of solving a specific problem of synthesis of control of a complex technical object based on the approximation of optimal trajectories is given. It is shown that such a control, obtained on the basis of machine learning, gives good results not only for the input data from the training set, but also not from it.

The concept of machine learning is widely known, but very often limited by its association with neural network technology. We are expanding the concept of machine learning to include a description of an unknown function in its formulation. Thus, a function can be specified and training is aimed only at finding parameters, as in neural networks, but you can also search for the structure of the function and its parameters. This became possible with the advent of symbolic regression methods. The complexity of these methods lies in the need to organize search in a space in which there is no metric. This greatly complicates the solution of the problem of finding the required structure of the function. This complexity opens up a wide field of research. One of the ways to solve this problem is to use the principle of small variations of the basic solution indicated in the article. This

approach allows you to concentrate the search for a solution around a basic solution based on the developer's experience or intuition. This approach also requires further study.

**Author Contributions:** Conceptualization, A.D. and E.S.; methodology, A.D., E.S. and S.K.; software, S.K.; validation, E.S. and G.D.; formal analysis, A.D., G.D.; investigation, S.K.; data curation, S.K.; writing—original draft preparation, A.D., E.S. and S.K.; writing—review and editing, E.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is supported by the Ministry of Science and Higher Education of the Russian Federation, project No. 075-15-2020-799.

**Conflicts of Interest:** The authors declare no conflict of interest.
