Calculation Of Numerical Green's Function Based On Neural Network

In electromagnetic theory,Green's function represents the response(field intensity)due to a point source of the unit intensity under certain boundary conditions.Commonly used Green's functions with analytical form can only be obtained under certain boundary conditions,such as free-space Green's function or half-space Green's function.For more general complex boundary conditions,Green's function can only be solved by numerical methods,namely numerical Green's function.Traditional methods for solving numerical Green's function include finite element method,finite-difference time-domain method,and method of moment.Numerical Green's functions solved by these numerical methods are generally expressed in matrix form,and their computational complexity and required storage resources are huge.Therefore,seeking a faster and more convenient numerical Green's function solution is very important for the development of electromagnetic computing.This thesis presents an algorithm for solving numerical Green's function based on neural network.This method utilizes the powerful function fitting ability of the neural network to approximate and fit the implicit numerical Green's function,which links the field-source coordinate with the point source response in the form of a composite function.According to the complexity of neural network's structure and numerical Green's function's training data form from simple to complex,our work can be divided into three stages.First,we studied the computation of numerical Green's function with classical definition based on the fully connected neural network.This algorithm directly uses the field value solved by the method of moment as the output of the dataset.Hence,the dataset structure used in this algorithm is simple.By training the fully connected neural network,we can obtain the composite function which connects the source-field position and the field value.And this function could represent the numerical Green's function.Second,in order to obtain the numerical Green's function that can be applied to the electric field integral equation,this work further studies the neural-network-accelerated numerical Green's function.This method uses a certain matrix technique to separate the known direct term of the numerical Green's function from the unknown scattering term,and utilizes a neural network to fit the relationship between the field-source position and the scattering term,and finally the neural-network-accelerated numerical Green's function is applied to the solution of scattering problems.As this method extracts the unknown part,the training data is more in line with the requirements of independent and identical distribution,so that the fitting function of the neural network can be better played.Finally,this paper proposes a convolutional neural network based on multilayer fast multipole method.This method determines the convolution kernel parameters related to multipole-expansion-to-multipole-expansion operation,local-expansion-to-local operation and multipole-expansion-to-local-expansion operation through the analogy of the similarities between the multilayer fast multipole algorithm and the convolution algorithm,as well as the convolution operation hyperparameters related to multipole expansion,local expansion and neighbor-group operation.And field images are used as training data to optimize the specific parameters of these three convolution kernels.When using this convolutional network to solve the numerical Green's function,we use the field image excited by the point source on the scatterer as the input tensor.The output of the network is the current distribution of the scatterer due to point source excitation,and the field generated by this equivalent current in space is the numerical Green's function to be calculated.As this method makes full use of the prior knowledge of computational electromagnetics,the number of parameters of the neural network constructed here is greatly reduced compared with the aforementioned fully connected neural network,and the number of training datasets and optimization iterations can also be reduced accordingly.This provides ideas for solving larger-scale problems in the future.