Summary: We suggest an approach to speed up the Gauss-Newton solution of inverse partial differential equation problems by minimizing the number of forward problem calls. The acceleration is based on effective incorporation of the information from the previous iterations via a reduced-order model (ROM). It is designed with the help of Galerkin and pseudo-Galerkin methods for selfadjoint and complex symmetric problems respectively. The constructed ROM generates effective multivariate rational interpolation matching the forward solutions and the Jacobians from the previous iterations. Numerical examples for the inverse conductivity problem for the 3D Maxwell system show significant accelerations.