A new approach to solve inverse problems: Combination of model-based solving and example-based learning

Inverse problem, which is one of the basic forms of mathematical problems, exists in the science, engineering and technology extensively. Traditional inverse problems are resolved through solving mathematical models related to problems (such as partial differential equations, and variation problems). A great deal of approximations are unavoidably employed during both the modeling and model solving processes, so it is hard to achieve an exact solution. Owing to this point, these inverse problems are universally recognized as hard problems. However, it is noticed that, besides models, a large number of examples exist for many inverse problems. Inspired by this condition, a new approach for solving the inverse problems which combines model solving and example-based learning is proposed in this work. Compressed sensing magnetic resonance imaging (CS-MRI) is taken as an example to illustrate how to incorporate compressed sensing model solving and deep learning based on example-based learning, which forms a totally new method to solve CS-MRI problem. Applications demonstrate that the new method is not only feasible and efficient, but also has high generalization.