| The inverse source problem has important applications in many fields of science and technology,such as dangerous chemical detection,medical imaging,photoacoustic and thermoacoustic tomography,brain imaging,gesture recognition of artificial intelligence and so on.It has been a longstanding difficulty to accurately reconstruct small disturbances on the source boundary.In this thesis,we are concerned with the reconstruction of acoustic sources which may exist a small high-curvature component based on machine learning method and the Fourier method.The main innovations of this thesis are the analysis of quantitative relationship between highfrequency Fourier coefficients and high curvature boundary and establishing the geometry generation model by combining neural network with the improved Fourier method.Firstly,we introduce the research status of inverse source problem and the development process of introducing machine learning method in statistics into inverse problem.Secondly,the basic knowledge of inverse source problem and Fourier method are introduced.Then,we investigate the relationship between high frequency Fourier coefficients and high-curvature boundary.Our finding shows that the decaying speed of Fourier coefficients is various in different directions and depends on curvature.Therefore,by selecting appropriate Fourier coefficients,we can enhance the reconstruction of high-curvature boundary.We also establish the stability of our method.This analysis can be a new perspective for super-resolution reconstruction of the small perturbation.Moreover,we combine neural network with the improved Fourier method to establish a geometry shape generation model,which can reconstruct shape of the object by a small number of characteristic parameters.Finally,numerical experiments are carried out to illustrate effectiveness of the proposed method.Numerical results show that this method can produce satisfactory reconstructions. |
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