Neural Network Method For Solving Two Kinds Of Delay Differential-algebraic Equations

Delay differential-algebraic equations(DDAEs)often arise in many fields of scientific engineering,such as circuit design,mechanical systems,power systems,control theory,multibody control systems,biological systems,atomic systems,fluid mechanics,chemical engineering,etc.Due to its special structure,researchers have poured a lot of effort to solve DDAEs.A lot of work has been done on the structure and properties of DDAEs.It is difficult to carry out numerical simulation due to the influence of time delay and the constraints of algebraic conditions.Neural networks have many advantages such as large-scale parallel processing,distributed storage,adaptiveness,fault tolerance,and redundancy,which have developed rapidly in recent years.Thier research covers various fields,and many domestic and foreign scholars have applied them to solve various types of differential equations.So far,there is few research work on neural networks to solve DDAEs.Legendre neural network combined with the optimization algorithm of extreme learning machine(ELM)is used to solve linear DDAEs.First,the solution interval is divided into multiple subintervals.In each subinterval,the output weight of the neural network is optimized by using ELM to obtain the optimal numerical solutions,and then the numerical solutions on the entire interval is obtained.The convergence of the method is proved in this paper.Numerical examples verify the effectiveness of the method.Tanh neural network combined with Adam optimization algorithm is used to solve nonlinear DDAEs.Similarly,the solution interval is divided into multiple subintervals.The loss function of the original equations on each subinterval is calculated,and then the Adam optimization algorithm is applied to minimize the objective function.Numerical examples verify the effectiveness of the method.