Damage detection in composite materials is a critical challenge in structural health monitoring. We investigate how to identify and quantify damage using a combination of mathematical models and measurement data. The governing physics are described by a second-order mechanical system which is derived by discretizing the underlying equation of motion. Damage is introduced through a parametrization (including e.g. damage size, Young's modulus reduction, damage location) and appears as parametrized stiffness matrix in the model. Estimating these parameters requires a huge number of high-dimensional simulations. To overcome this, we employ model reduction techniques. As a first step, we compare classical intrusive reduction approaches with data-driven non-intrusive techniques on a simplified model of a wave equation. However, for practical applications full access to the system matrices is often not possible. Therefore, we continue with studying non-intrusive methods, in particular operator inference, to infer damage parameters from observed system responses. This work is supported by the DFG FOR 3022, project number 418311604 and is joint work with Saddam Hijazi.