Applications of Topological Derivatives and Neural Networks for Inverse Problems

Numerical method of identification for small circular openings in the domain of integration of an elliptic equation is presented. The method combines the asymptotic analysis of PDE’s with an application of neural networks. The asymptotic analysis is performed in singularly perturbed geometrical domains with the imperfections in form of small voids and results in the form of the so-called topological derivatives of observation functionals for the inverse problem under study. Neural networks are used in order to find the mapping which associates to the observation shape functionals the conditional expectation of the size and location of the imperfections. The observation is given by a finite number of shape functionals. The approximation of the shape functionals by using the topological derivatives is used to prepare the training data for the learning process of an artificial neural network. Numerical results of the computations are presented and the probabilistic error analysis of such an identification method of the holes by neural network is performed.