Neural Network Method Based On Stochastic Optimization For Solving Partial Differential Algebraic Equations

Partial differential algebraic equations(PDAEs)often appear in mathematical modeling and physics problems,PDAEs are widely used in multibody mechanics,spacecraft control,and incompressible fluid mechanics.Algebraic constraints bring difficulties to solve PDAEs.Therefore,it is significance to study the numerical methods and theories of PDAEs.In recent years,due to the improvement of computer capabilities,especially the continuous improvement of hardware resources(GPU,TPU),neural networks,especially deep neural networks,have achieved rapid development and progress.Based on the good adaptability and similarity of neural networks,our article uses many different types of deep neural networks for solving PDAEs.The main contents of the article are as follows:The first chapter is the preface,which mainly introduces the research background,current status and significance of PDAEs and neural network and the research progress of the differential equations solved by neural networks.The second chapter is preliminary knowledge,which mainly introduces the specific structure and propagation process of neural networks,ordinary activation functions and typical optimization methods.The third chapter introduces the neural networks for solving linear PDAEs.Firstly,the neural network structure is constructed.Determine the dimensions of the input data and output data according to the number of independent variables in the equation and the the number of functions to be sought.Then,determine the number of hidden layers and neurons,select the tanh function(other types of activation functions can be selected)as the activation function,use fully connected network method to construct the neural network structure.Next,the loss function of the neural network is determined according to the equation.The loss function consists of the internal loss of the equation and the loss of initial conditions and boundaries conditions.Secondly,after the defined area is divided by a uniform grid,using the internal data points of the equation to construct the internal loss function of the equation and the data points of the initial conditions and boundary conditions to construct the initial boundary loss function.Based on the above analysis,our goal is to minimize the loss function.The optimization algorithm is used to obtain the parameters of the neural network,thereby,the approximate solution of PDAEs is obtained.Finally,select the sigmoid,relu,eluas activation functions combined with optimization algorithms batch gradient descent,and momentum and random optimization algorithms root mean square prop and adaptive moment estimation separately for numerical simulation.The main parts of the experiment are as follows: 1.In order to study the effect of the neural network structure on the results,the results under different layers of hidden layers,different numbers of neurons and different activation function are compared;2.In order to study the effects of external conditions on the results,the results under different optimization methods and different input data are compared.The fourth chapter is to use the neural network to solve the nonlinear PDAEs.On the basis of Chapter 3,a new loss function according to the form of nonlinear PDAEs is defined.Finally,carry out numerical simulations similar to Chapter 3 and the availability of neural network method with a new activation function sinh is considered,the numerical test shows that this type of method also applies to nonlinear PDAEs effectively.The fifth chapter is a summary,which mainly summarizes the main research results,points out the highlights and deficiencies of this article,and discusses further research directions.