Machine Learning For Inverse Scattering Problems

The objective of the inverse scattering problem is to determine the geometric or physical parameters of scatterers through the observaed wavefield data.Such inverse problems have broad applications in fields such as medical imaging,geophysics,materials design,etc.With the developments in artificial intelligence and big data,machine learning has made significant breakthroughs in various disciplines and applications in recent years.This paper investigates the application of machine learning to solving inverse scattering problems from the perspectives of data and model.Firstly,we consider measurement data from a single incident direction concerning the inverse scattering problem of obstacles and medium.From a theoretical perspective,the uniqueness of such data for the inversion of arbitrary scatterers remains an open problem.Here,we only numerically investigate inversion algorithms.To approach practical scenarios,we separately consider far-field data,phaseless far-field data,radar cross-section data,and finite radar cross-section data.Due to incomplete data,this inverse problem is nonlinear and ill-posed.Therefore,it poses significant challenges and holds crucial applications in practice.In this thesis,we take the advantages of machine learning by integrating fully connected neural networks with data-driven method into the inverse scattering problem.This method is not only easy to implement but also exhibits promising inversion results for severely incomplete data.Subsequently,we consider the inverse scattering problem of obstacles based on limited-aperture observation data.Due to measurement techniques and inherent properties of the problem,obtaining full-aperture data is often challenging,then this problem is more practical.Classical sampling methods exhibit local characteristics in the inversion of limited-aperture observation data.We propose a two-step inversion algorithm by combining convolutional neural networks with sampling methods.Firstly,we design a specialized convolutional neural network along with a regularizationbased loss function to recover limited-aperture data,thus approximating the mapping from limited-aperture data to full-aperture data in a data-driven method.Then,we employ classical sampling methods to reconstruct the obstacle from the recovered full-aperture data.This approach not only closely approximates the inversion results obtained from real full-aperture data but also yields promising inversion outcomes for phaseless data from limited apertures and even sparser limited-aperture data.Finally,we consider the inverse scattering problem of obstacles in acoustic waveguides.As the two aforementioned methods require a large number of training samples for constructing approximating mappings using neural networks.Based on Bayesian inference methods,we consider reconstructing unknown scatterer shapes.Firstly,we establish stability results of the posterior probability measure in terms of the f-divergence.Then,to expedite the sampling process of Markov chain Monte Carlo algorithms while ensuring sample accuracy,we construct a surrogate model combining neural networks with waveguide modes and introduce an adaptive multifidelity strategy.Finally,we provide error approximation results in terms of the f-divergence.Numerical experiments show that our algorithm not only achieves high-precision reconstruction but also substantially reduces computational costs.