Numerical Solution of Inverse Problems of Wave Dynamics in Heterogeneous Media with Convolutional Neural Networks

The chapter considers the problem of restoring the shape of the boundary between two elastic mediums with different rheological properties. The problem has many variations. This chapter concentrates on the application to the medical ultrasound aberrations. The problem statement includes two acoustically contrast materials: the curved layer of acoustically rigid material on the surface and the main volume of acoustically soft material. The direct problem of the propagation of the ultrasonic pulse in the medium is solved numerically using the discontinuous Galerkin method. The inverse problem is solved using neural networks. The proposed method is based on the assumption that the continuous observations of wave amplitude and velocity on the surface of the medium could also be treated as images, and the initial problem can be reduced to the optimization problem. The neural networks are trained using the data from multiple numerical solutions. During the testing phase, the networks restored the shape of the boundary using the data from the surface. Three different neural network architectures were tested and compared.