Inversion techniques aim to infer unknown quantities from observable data. Well-established variational methods, widely used in image processing, utilize iterative approaches. These data-agnostic methods yield great success but suffer computation challenges, including ill-posedness, uncertainty quantification, and scalability. Contrary, modern data-driven approaches seek reconstructions directly from available training data with many computational advantages but a loss of physical properties and interpretability. In this talk, we discuss new developments, that combine the strength of both approaches seeking a balance between computational burden and interpretability.