Neural network based solution to inverse problems

The well-posedness of problems is not always guaranteed in inverse problems, unlike in forward problems. Thus, a number of methods for giving well-posedness have been studied in mathematical fields. In the field of neural networks, the network inversion method for solving inverse problems was proposed; it is useful but does not remove the ill-posedness of inverse problems. To overcome the difficulty, we propose the answer-in-weights scheme to provide the network with a priori given knowledge. We compare the performance of answer-in-weights network with the one of an inversion network in solving the ill-posed inverse problem arising in the Fredholm integral equation of the first kind. Furthermore, we compare the expression of the a priori knowledge inherent to the problem, by using two kinds of models.