In the absence of analytical inversion formulas, reconstructions in inverse problems are traditionally achieved through costly computational methods (such as those based on the classical least-squares inversion). In recent years, various machine learning methods have been proposed as new computational tools to solve inverse problems. In this talk, we will discuss some of our recent understandings of how to couple machine learning methods and classical inversion techniques for inverse coefficient problems for PDEs. While most of the results are computational, we do attempt to provide some theoretical justifications when possible.