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We consider the inverse problem of determining the geometry of penetrableobjects from scattering data generated by one incident wave at a fixedfrequency. We first study an orthogonality sampling type method which is fast,simple to implement, and robust against noise in the data. This sampling methodhas a new imaging functional that is applicable to data measured in near fieldor far field regions. The resolution analysis of the imaging functional isanalyzed where the explicit decay rate of the functional is established. Aconnection with the orthogonality sampling method by Potthast is also studied.The sampling method is then combined with a deep neural network to solve theinverse scattering problem. This combined method can be understood as a networkusing the image computed by the sampling method for the first layer andfollowed by the U-net architecture for the rest of the layers. The fastcomputation and the knowledge from the results of the sampling method helpspeed up the training of the network. The combination leads to a significantimprovement in the reconstruction results initially obtained by the samplingmethod. The combined method is also able to invert some limited apertureexperimental data without any additional transfer training.

Sampling type method combined with deep learning for inverse scattering with one incident wave