Structure-Driven Inversion: A New Paradigm for Solving Inverse Problems

RESUMO

Inverse problems are predominantly solved within the optimization-driven paradigm, which formulates the problem as objective-function minimization and approaches the solution by iterative search. Though universal, it suffers from limitations in efficiency, interpretability, and multi-parameter decoupling. This paper proposes a new paradigm---Structure-Driven Inversion (SDI). Instead of iterative search, SDI identifies and exploits the intrinsic structure of the problem to construct a solution method. It has two types of structure---mathematical structure and physical structure---and two corresponding driving inversion types: Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). The current representative method of MSDI is Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII), which constructs a pseudo-inverse in the spectral domain via unitary diagonalization. The current representative methods of PSDI are Physical Structure-Driven Waveform Inversion Imaging (PSDWII), which performs inversion through virtual-source projection, and Physical Structure-Driven Back-Propagation (PSDBP), which implements three structural projections for deep neural networks via physical-system analogy. SDI is not a rejection but a complement and extension of Optimization-Driven Inversion (ODI). To the best of the author's knowledge, no existing study has systematically presented structure-driven inversion as an independent paradigm; this paper aims to fill that gap.