The Stability Analysis And Application Of Neural Networks

The neural network is a type of network model with strong information handling capacity,which has been broadly applied in video analysis,machine translation,image classification and many other fields.The success of network model attributes to the nonlinear network layers of a large number of the complex structures,which can extract feature information from input data at various abstraction levels.However,because the neural network has many parameters and highly complex structures,it’s interpretability is poor.It is a hot spot for analyzing the internal structure of the neural network by complete mathematical theories in the machine learning.Hence,the focus of this paper is to interpret the neural networks as the discrete differential equations and analyze the stability of neural networks according to the stability theories of differential equations.Considering the connection between neural networks and differential equations,this paper proposes a conditionally stable network unit called the GUEM and a stable network unit called the RUEM.Firstly,this paper briefly introduces the relationship between the recurrent neural network and the ordinary differential equation,and designs two network units respectively called the GUEM and the RUEM according to the gated thought of recurrent neural networks and the forward Euler’s method of differential equations.Secondly,the judgment theorems of stability are used to analyze the stability of the two network units without input variable.The numerical simulations are carried out in two dimensional space and three dimensional space.Finally,by the GUEM and the RUEM,this paper constructs two single-layer recurrent neural networks from sequence to sequence respectively called the GUEM-RNN and the RUEM-RNN.In order to verify the effectiveness of the GUEM-RNN and the RUEM-RNN,this paper respectively applies the two recurrent neural networks to solve the inverse scattering problem of obstacles.This paper considers the far-field data and the Fourier coefficients of truncated boundary curve equation of obstacle as the input sequence and output sequence of the neural network.The weight matrices and biases are updated by the mean square loss function and Adam optimization algorithm,so as to accurately reconstruct the shape of obstacles.The numerical experiments show that the GUEM-RNN and the RUEM-RNN can effectively tackle the inverse scattering problem and have many merits such as fast convergence speed.Moreover,even in the situation of far-field data of lower noise levels,the two recurrent neural networks also have the good reconstruction effects.