In recent decades, the topic of depicting network structures from output variable data, i.e., the inverse problem, turns to be a key issue in wide interdisciplinary areas, in particular, in biological and social fields. Noise inevitably exists in practical dynamic networks, and the output data are often generated via interplay between noise and network structures. The essential difficulty to solve the inverse problem is how to extract information of node links in networks under unknown and possibly strong noise. In this paper, based on the idea that the output variable data contain information not only for network topology but also for noise, we propose a method to deal with this problem, incorporating three crucial ingredients: Computing multiple matrices to extract as much as possible information on network topology and noise statistics; making a systematical matrix algebraic computation to obtain equations closed for network inference; using an effective iteration algorithm to solve the resulting nonlinear matrix equations. The above theory is established in an accurate and closed form, numerical computations convincingly verify the validity of theoretical analysis, and the possible applications in practical inverse problems are emphasized.