Neural network augmented inverse problems for PDEs

In this paper we study the classical coefficient approximation problem for partial differential equations (PDEs). The inverse problem consists of determining the coefficient(s) of a PDE given more or less noisy measurements of its solution. A typical example is the heat distribution in a material with unknown thermal conductivity. Given measurements of the temperature at certain locations, we are to estimate the thermal conductivity of the material by solving the inverse problem for the stationary heat equation. The problem is of both practical and theoretical interest. From a practical point of view, the governing equation for some physical process is often known, but the material, electrical, or other properties are not. From a theoretical point of view, the inverse problem is a challenging often ill-posed problem in the sense of Hadamard. Inverse problems have been studied for a long time, starting with Levenberg [24] in the 40's, Marquardt [29] and Kac [20] in the 60's, to being popularized by Tikhonov [36] in the 70's by his work on regularization techniques.