Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator

An innovative numerical method based on an artificial neural network is presented in order to solve an inverse problem associated with the calculation of the Dirichlet eigenvalues of the anisotropic Laplace operator. Using a set of predefined eigenvalues obtained by solving repeatedly the direct problem, a radial basis neural network is designed with the purpose to find the appropriate components of the anisotropy matrix, related to the Laplace operator, and thus solving the associated inverse problem. The finite element method is used to solve the direct problem and to create the training set for the first radial basis neural network. A nonlinear map of the Dirichlet eigenvalues as a function of the anisotropy matrix is then obtained. This nonlinear relationship is later inverted and refined, by training a second radial basis neural network, solving the aforementioned inverse problem. Some numerical examples are presented to prove the effectiveness of the introduced method.