An innovative numerical method based on a neural network approach is used to solve inverse problems involving the Dirichlet eigenfrequencies for different partial differential operators in bounded domains filled with solid composite materials. The inhomogeneity of the investigated materials is characterized by a vector that is designed to control the constituent mixture of solid homogeneous materials that compose these materials. Using the finite element method, we create a training set for a forward artificial neural network, solving the forward problem. A forward nonlinear map of the Dirichlet eigenfrequencies as a function of the vector design parameter is then obtained. This forward relationship is inverted and applied to obtain a training set for an inverse radial basis neural network, solving the aforementioned inverse problem. Numerical examples that demonstrate the applicability of this methodology are presented.