Using recently proposed tensor train format for the representation of multi-dimensional dense arrays (tensors) we develop a fast interpolation method to approximate the given tensor by using only a small number of its elements. The algorithm is based on DMRG scheme, known among the quantum chemistry society. It is modified to make an interpolation on the adaptive set of tensor elements. The latter is selected using the maximum-volume principle which was previously used for the cross approximation schemes for matrices and 3-tensors. The numerical examples includes the interpolation of one- and many-dimensional functions on the uniform grids.