In the known methods of symbolical regression by search of the solution with the help of a genetic algorithm, there is a problem of crossover. Genetic programming performs a crossover only in certain points. Grammatical evolution often corrects a code after a crossover. Other methods of symbolical regression use excess elements in a code for elimination of this shortcoming. The work presents a new method of symbolic regression on base of binary computing trees. The method has no problems with a crossover. Method use a coding in the form of a set of integer numbers like analytic programming. The work describes the new method and some examples of codding for mathematical expressions.

This study takes a look into the application of the Naive Bayes machine learning algorithm to enhance the accuracy of Intrusion Detection Systems (IDS). The primary focus is to assess the algorithm's performance in detecting various types of network attacks, particularly Denial of Service (DoS) attacks. This research proposes using Naive Bayes to improve intrusion detection systems that struggle to keep pace with evolving cyber threats. This study evaluated the efficiency scores of the Naive Bayes classifying model for two different dependency scenarios and identified strong and weak properties of this model. The Naive Bayes classifier demonstrated satisfactory results in detecting network intrusions, especially in binary classification scenarios where the goal is to distinguish normative and malicious traffic due to its simplicity and efficiency. However, its performance declined in multi-class classification tasks, where multiple types of attacks need to be differentiated. The study also highlighted the importance of data quality and quantity in training machine learning models because of the impact of those parameters on the model efficiency. The USB-IDS-1 dataset, while useful, has limitations in terms of the variety of attacks. Using datasets with a wider range of attack types could significantly improve the accuracy of IDS. The findings of this research can be applied to such domains as network security, cybersecurity, and data science. The Naive Bayes classifier can be integrated into IDS systems to enhance their ability to detect and respond to cyber threats. However, it is essential to consider the limitations of the algorithm and the specific conditions of its environment. To maximize the effectiveness of the Naive Bayes classifier, it could be promising to optimize and normalize the data to improve the accuracy of the model and combine Naive Bayes with the other machine learning algorithms to address its limitations

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object’s control system.

The extended optimal control problem is considered. It is necessary to find an optimal control function, that not only provides the achievement of terminal state with optimal value of the given quality criterion, but also is implemented in the control system of a real object. It means, that the control function should depend on the state space vector, and the optimal solution should keep optimality property at small perturbations of the found solution. To solve this problem machine learning control by symbolic regression is used. In the extended optimal control problem, the problem statement of stabilization system synthesis for movement along the optimal trajectory is included. Synthesis problem is solved by the network operator method. In the synthesis problem a domain of initial conditions is considered instead of one point of initial state. It provides less sensitivity of found solution to perturbations of initial states. An example of solving the extended optimal control problem with complex phase constraints in the form of bottleneck for four quadcopters is presented.

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper describes a stabilisation system synthesis for motion along a trajectory specified in time and along a trajectory specified in the form of a manifold in a state space. In order to construct a stabilisation system, it is necessary to determine a distance between an object and the given trajectory at each moment in time. For trajectories that are not given in time, the determination of this distance can be ambiguous. An object may be exactly on a trajectory but at a different time. This paper proposes some approaches to solve the problem. One of the approaches is to transform a given trajectory in a state subspace into a trajectory given in time. A description of a universal method to perform this transformation is presented. In order to solve the synthesis problem automatically, without having to analyse the mathematical model of the control object, it is suggested that machine learning control by symbolic regression is used. In computational experiments, examples of stabilisation system syntheses for quadcopter motion along a given spatial trajectory are presented.

The paper presents a universal numerical method of symbolic regression for machine learning control, which allows to obtain solutions for any mathematical model of a control object.