Solving the inverse problem to find a hidden object

I was listening to StarTalk and Neil deGrasse Tyson told the story of the discovery of Neptune, which first happened mathematically and then observationally! Essentially, the effect of Neptune on the trajectories of other planets caused discrepancies that shouldn't have been there. This clued-in two mathematicians of the 19th century, John C. Adams and Urbain Le Verrier , who looked into what…