Structure-Driven Methodology: An Emerging Cross-Domain Paradigm

This article proposes and articulates structure-driven methodology as an emerging cross-domain paradigm. In contrast to the objective-driven paradigm that has dominated inverse-problem solving for over two centuries, structure-driven methodology advocates identifying and exploiting the intrinsic structure of a problem to construct the solution method directly, rather than approaching the solution through iterative search. Drawing on examples from the Liouville-variant Goldbach conjecture, the three-dimensional Kakeya conjecture, Noether’s theorem, and the author’s nearly four decades of geophysical inversion practice, the article illustrates the common logic of structure-driven thinking in number theory, pure mathematics, physics, and seismic imaging. The author also systematically proposes the Structure-Driven Inversion (SDI) paradigm, comprising Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). Structure-driven methodology does not negate objective-driven methodology but complements it: objective-driven is a general search strategy, whereas structure-driven is a custom direct strategy. The article concludes that the significance of this shift lies not only in efficiency gains but also in cognitive deepening—when facing complex problems, the first question is no longer how to solve the problem faster'' but what is the structure of the problem.‘’