METHODS OF SOLVING OPTIMAL SOLUTIONS OF MATHEMATICAL PROBLEMS WITH ARTIFICIAL INTELLIGENCE METHODS

Аннотация

В статье показано, что решение математических задач может быть более эффективно реализовано с использованием искусственного интеллекта (ИИ), чем традиционными методами. Искусственный интеллект, в частности нейронные сети и методы глубокого обучения (Deep Learning), значительно упрощают и ускоряют процесс решения математических и статистических задач. В данной работе рассматриваются применение, возможности и практические подходы методов искусственного интеллекта при решении математических задач. Ключевые слова: Искусственный интеллект, нейронные сети, глубокое обучение (Deep Learning), регрессия, классификация, оптимизация, моделирование, дифференциальные уравнения, машинное обучение, Выпуск журнала №-22 Часть–3_Март –2025 248 MODERN EDUCATION AND DEVELOPMENT генетические алгоритмы, математическое моделирование, функции ошибок, физико-информированные нейронные сети (PINN). Complex problems, such as regression, classification, and optimization, can be solved using neural networks, evolutionary algorithms, genetic algorithms, and other machine learning methods. These methods help to identify ambiguous and complex relationships between input and output data. Artificial intelligence methods are also effectively used in complex problems such as simulation, optimization, and differential equations. Methods for solving mathematical problems using artificial intelligence are currently very popular and are used in many different fields. These methods can be divided into the following main areas: • Genetic algorithms and simulation methods can be effective for solving mathematical problems. For example, in complex optimization problems, the best solution can be found using genetic algorithms or simulation methods (Monte Carlo simulation). Simulation and optimization are widely used methods in the fields of artificial intelligence and mathematics, which help in solving many real-world problems. Let's take a closer look at each of these two concepts: Simulation is the process of creating a real or theoretical model of a system or process and testing its behavior or performance on a computer. This method is used to understand systems or processes, predict them, or make decisions. • Monte Carlo Simulation: This method allows you to test the behavior of a system using probability distributions. Monte Carlo simulation is a method for modeling the behavior of a system using random numbers. It is used, for example, to model or predict random processes. • Agent-based simulation: In this method, the system is represented by different agents (such as people, machines, or other objects). Each agent learns its own behavior to create the overall behavior of the system. The agents interact with each other and simulate the system. Выпуск журнала №-22 Часть–3_Март –2025 249 MODERN EDUCATION AND DEVELOPMENT • Dynamic simulation: This method aims to simulate how a system changes over time. For example, it is used to model economic or ecological systems. Differential equations and integrals can be used to analyze the change in a system over time. Helps to understand the behavior of complex systems. Allows you to study processes or problems that are difficult to test in practice. Allows you to quickly test many alternative solutions. Can be computationally and time-consuming. Models and approaches may not fully reflect reality. In some cases, the necessary data for creating a model may not be available. Optimization is the process of finding the best (optimal) solution for a system to achieve a specific goal. This process requires the use of various algorithms and mathematical models. Optimization is used in many areas, such as manufacturing, transportation, marketing, economics, and others. Linear Optimization. In this type of optimization, the objective function and constraints are linear. For example, it is used in the allocation of resources in production, minimizing costs, or maximizing profits. The simplex algorithm is one of the most popular methods for linear optimization. Nonlinear Optimization: If the objective function or constraints are nonlinear, then nonlinear optimization methods are used. Solutions to such optimization problems are found using gradient methods or heuristic methods. Integer Optimization: In this type of optimization, the variables must be integers. For example, it is used in logistics problems when choosing freight transport routes or in the allocation of resources in the production process. Genetic Algorithms: Genetic algorithms (GAs) are heuristic methods that mimic the processes of natural selection and heredity. These algorithms are useful in solving complex optimization problems because they help solve global optimization problems better than local optimization. Helps find the most efficient and economical solution to the problem. Allows for efficient allocation of resources and maximum benefit. Выпуск журнала №-22 Часть–3_Март –2025 250 MODERN EDUCATION AND DEVELOPMENT Effective in developing solution