The solution of fractional-order systems has been a complex problem for our research.Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive,which make...The solution of fractional-order systems has been a complex problem for our research.Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive,which makes it more difficult for our solution efficiency.The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems.Therefore,this study considers how to improve the accuracy and efficiency of the solution based on traditional methods.Finally,we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model(FODS-NAR).First,we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system.Second,we compare the FODS-NAR algorithm with the famous and good reservoir computing(RC)algorithms.We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters,and the residuals of the FODS-NAR algorithm are closer to 0.Consequently,we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems.In addition,we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.展开更多
Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties.In this paper,we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction ...Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties.In this paper,we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction for both the original system and the system driven by noise.This demonstrates that deep learning can be applied in dynamics of population.This is the first study that uses deep learning algorithms to predict Lotka-Volterra equations.Several numerical examples are presented to illustrate the performances of the proposed algorithm,including Predator nonlinear breeding and prey competition systems,one prey and two predator competition systems,and their respective systems.All the results suggest that the proposed algorithm is feasible and effective for predicting Lotka-Volterra equations.Furthermore,the influence of the optimizer on the algorithm is discussed in detail.These results indicate that the performance of the machine learning technique can be improved by constructing the neural networks appropriately.展开更多
基金supported by the National Natural Science Foundation of China(NNSFC)(Grant No.11902234)Natural Science Basic Research Program of Shaanxi(Program No.2020JQ-853)+1 种基金Shaanxi Provincial Department of Education Youth Innovation Team Scientific Research Project(Program No.22JP025)the Young Talents Development Support Program of Xi’an University of Finance and Economics.
文摘The solution of fractional-order systems has been a complex problem for our research.Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive,which makes it more difficult for our solution efficiency.The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems.Therefore,this study considers how to improve the accuracy and efficiency of the solution based on traditional methods.Finally,we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model(FODS-NAR).First,we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system.Second,we compare the FODS-NAR algorithm with the famous and good reservoir computing(RC)algorithms.We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters,and the residuals of the FODS-NAR algorithm are closer to 0.Consequently,we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems.In addition,we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.
基金supported by the National Natural Science Foundation of China(No.11902234)Natural Science Basic Research Program of Shaanxi(No.2020JQ-853)+2 种基金China(Xi’an)Silk Road Research Institute Scientific Research Project(No.2019ZD02)Shaanxi Provincial Department of Education Youth Innovation Team Scientific Research Project(No.22JP025)the Young Talents Development Support Program of Xi’an University of Finance and Economics。
文摘Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties.In this paper,we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction for both the original system and the system driven by noise.This demonstrates that deep learning can be applied in dynamics of population.This is the first study that uses deep learning algorithms to predict Lotka-Volterra equations.Several numerical examples are presented to illustrate the performances of the proposed algorithm,including Predator nonlinear breeding and prey competition systems,one prey and two predator competition systems,and their respective systems.All the results suggest that the proposed algorithm is feasible and effective for predicting Lotka-Volterra equations.Furthermore,the influence of the optimizer on the algorithm is discussed in detail.These results indicate that the performance of the machine learning technique can be improved by constructing the neural networks appropriately.
