Computation Of Periodic Solutions Of Delay Differential Equations Using Machine Learning

Delay differential equations are an important branch of functional differential equations.They are a class of ordinary differential equations that depend on the state of past time.Since the twentieth century,with the needs of various disciplines in the natural and social sciences,a large number of problems with time-delay dynamic systems have emerged.The study of solutions to time-delay differential equations has become the key to solving problems.There are many qualitative analysis of the solution of functional differential equations,such as the existence and uniqueness of solutions,stability,vibration,asymptotic and boundedness,etc.This article mainly studies the problem of solving periodic differential solutions of delay equations.Machine learning is the study of how computers simulate or implement human learning behaviors in order to acquire new knowledge or skills and reorganize existing knowledge structures to continuously improve their performance.It is the core of artificial intelligence and the fundamental way to make computers intelligent.The application of machine learning has spread to all areas of artificial intelligence,and machine learning algorithms are becoming more and more intelligent.Traditional optimization methods,such as Newton’s method,can be used to solve periodic solutions of delay differential equations.Traditional methods have high requirements on the differentiability of the system.At the same time,when the attracting domain of the periodic solution is small or the periodic solution is unstable,the calculation amount is very high.For this reason,intelligent optimization methods,such as genetic algorithm,particle swarm algorithm,neural network algorithm,are used to deal with the above problems,which expands the solution range and obtains higher accuracy.However,common intelligent methods are prone to fall into a local optimum,have randomness,and often take a long time to calculate.Therefore,this paper designs a local replacement model to be used in particle swarm optimization to enhance the performance of the optimization algorithm.Through the research and analysis of the previous work on the improvement of the particle swarm algorithm,this paper designs a local replacement model and embeds this model in the particle swarm algorithm to obtain a new improved particle swarm algorithm.And experiments verify that the performance of the new algorithm is more effective when dealing with optimization problems.The design of the partialreplacement model makes full use of the position information of the globally optimal particles in the particle swarm,and generates some new particles around the globally optimal particles.Based on the idea of "survival of the fittest" in the theory of biological evolution,the poorly adapted particles in the original population are replaced with the new particles produced.This replacement is local.The embedding of this model improves the diversity of the population particles,overcomes the shortcomings of the algorithm’s easy fall into local optimization,retains the advantages of the particle swarm algorithm,and converges faster in calculations.The results are more accurate and the number of calculations required and less.In summary,the main content of this article is to analyze the various solutions to the periodic solution of the delay differential equations,and to improve the particle swarm optimization algorithm,design a partial replacement model.And use the improved particle swarm algorithm to solve the periodic solution of the delay differential equation.First,the problem of solving periodic differential equations with delays is transformed into an optimization problem.Then,the improved particle swarm algorithm is used to solve the periodic solution of the delay differential equation.By comparing the results of several sets of numerical experiments,it can be found that the improved particle swarm algorithm has improved the calculation accuracy and convergence speed when solving the periodic solution of the delay differential equation.