Commonly used linear and nonlinear constitutive material models in deformation simulation contain many simplifications and only cover a tiny part of possible material behavior. In this work we propose a framework for learning customized models of deformable materials from example surface trajectories. The key idea is to iteratively improve a correction to a nominal model of the elastic and damping properties of the object, which allows new forward simulations with the learned correction to more accurately predict the behavior of a given soft object. Space-time optimization is employed to identify gentle control forces with which we extract necessary data for model inference and to finally encapsulate the material correction into a compact parametric form. Furthermore, a patch based position constraint is proposed to tackle the challenge of handling incomplete and noisy observations arising in real-world examples. We demonstrate the effectiveness of our method with a set of synthetic examples, as well with data captured from real world homogeneous elastic objects.