What is Schroedinger's equation an abstraction of? What details does it abstract from? How can it be that the omission of those details does not result in an inexact representation of the unabstracted, more complete, situation?
BTW, there are no exact solutions to the Schroedinger equation of most interesting systems: we only get numerical approximations to the values of interest from it.
Some of those numerical solutions were really important historically and lead to the further development of numerical techniques.
Back to main thread: I'm thinking of Lakatos' Proofs and Refutations, itself a play on Popper's Conjectures and Refutations, designed itself to escape from 'induction' based thinking, e.g. Hans Reichenbach
BTW, there are no exact solutions to the Schroedinger equation of most interesting systems: we only get numerical approximations to the values of interest from it.