We consider a new type of deep neural network developed to solve nonlinear inverse problems. In particular, we consider inverse problems for a wave equation where one want to determine an unknown wave speed from the boundary measurements. In particular, we consider the model where the wave propagation is governed by the linear acoustic wave equation on an interval. A novel feature of the studied neural network is that the data itself form layers in the network. This corresponds to the fact that data for the inverse problem is a linear operator that maps the boundary source to the boundary value of the wave that is reflected from the unknown medium. Even though the wave equation modelling the waves is linear, the inverse problem of finding the coefficients of this equation is non-linear. Using the classical theory of inverse problems we design a neural network architecture to solve the inverse problem of finding the unknown wave speed. This makes it possible to rigorously analyze the properties of the neural network.
For inverse problems, the main theoretical questions concern uniqueness, range characterisation, stability and the regularisation strategies for the inverse problems. We will discuss the question when a solution algorithm generalises from the training data, that is, when the solution algorithm trained with a finite number of samples can solve the problem with new inputs that are not contained in the training data. This can viewed as a new question for classical inverse problems that takes its motivation from machine learning.
The results are done in collaboration with Christopher A. Wong and Maarten de Hoop.
References: M. de Hoop, M. Lassas, C. Wong: Deep learning architectures for nonlinear operator functions and nonlinear inverse problems. Preprint: arXiv:1912.11090