Approximating the Electrical Impedance Tomography Forward Problem with Graph Neural Networks

Electrical impedance tomography is a differential tomography method where current is injected into a domain and its interior distribution of electrical properties are inferred from measurements of electric potential around the boundary of the domain. Within the context of this imaging method the forward problem describes a situation where we are trying to deduce voltage measurements on a boundary of a domain given the conductivity distribution of the interior and current injected into the domain through the boundary. Traditionally the problem has been solved either analytically or by using numerical methods like the finite element method. Analytical solutions have the benefit that they are efficient, but at the same time have limited practical use as solutions exist only for a small number of idealized geometries. In contrast, while numerical methods provide a way to represent arbitrary geometries, they are computationally more demanding. Many proposed applications for electrical impedance tomography rely on the method's ability to construct images quickly which in turn requires efficient reconstruction algorithms. While existing methods can achieve near real time speeds, exploring and expanding ways of solving the problem even more efficiently, possibly overcoming weaknesses of previous methods, can allow for more practical uses for the method. Graph neural networks provide a computationally efficient way of approximating partial differential equations that is accurate, mesh invariant and can be applied to arbitrary geometries. Due to these properties neural network solutions show promise as alternative methods of solving problems related to electrical impedance tomography. In this thesis we discuss the mathematical foundation of graph neural network approximations of solutions to the electrical impedance tomography forward problem and demonstrate through experiments that these networks are indeed capable of such approximations. We also highlight some beneficial properties of graph neural network solutions as our network is able to converge to an arguably general solution with only a relatively small training data set. Using only 200 samples with constant conductivity distributions, the network is able to approximate voltage distributions of meshes with spherical inclusions