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Abstract. Numerical method for identification of imperfections is devised for elliptic spectral problems. The neural networks are employed for numerical solution. The topological derivatives of eigenvalues are used in the learning procedure of the neural networks. The topological derivatives of eigenvalues are determined by the methods of asymptotic analysis in singularly perturbed geometrical domains. The convergence of the numerical method in a probabilistic setting is analysed. The method is presented for the identification of small singular perturbations of the boundary of geometrical domain, however the framework is general and can be used for numerical solutions of inverse problems in the presence of small imperfections in the interior of the domain. Some numerical results are given for elliptic spectral problem in two spatial dimensions.

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Abstract. 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 socalled 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.