An inverse problem is the process of calculating the hidden causes or factors that produced a set of observed effects. This is the direct opposite of a forward problem, which uses known causes and physical laws to predict future states or effects. Inverse problems are fundamental across scientific disciplines because they allow researchers to determine parameters that cannot be directly observed, effectively revealing hidden mechanisms from observable phenomena.

A defining characteristic of most inverse problems is that they are mathematically "ill-posed" according to Jacques Hadamard's criteria. This means they often violate at least one of three conditions: the existence of a solution, the uniqueness of the solution, or the continuous stability of the solution. The lack of stability is particularly challenging; because measurement operato ...