DEQ-Solver: A Posterior Score-based Deep Equilibrium ODE Solver for Inverse Problems in Medical Imaging

Score-based Generative Models have emerged as the dominant generative paradigm. Owing to their impressive generative performance and high interpretability, they have also proven to be a compelling generative prior for solving inverse problems in medical imaging. Nevertheless, incorporating the posterior distribution derived from the physical imaging model into the sampling process continues to be a nontrivial issue. Furthermore, their prohibitive sampling cost remains a major challenge. To overcome the above limitations, we propose a posterior score-based Deep Equilibrium Solver (DEQ-Solver). It constructs a measurement-conditioned neural vector field that directly estimates the posterior score. Based on the implicit backward Euler formulation, the resulting reverse-time dynamics can be cast as a global fixed-point problem. By solving for the global equilibrium, we can jointly drive all temporal states along the entire reverse-time trajectory. Therefore, our method improves the efficiency of solving medical imaging inverse problems without compromising reconstruction quality. Across various medical imaging modalities, DEQ-Solver achieves superior reconstruction quality with substantially faster sampling, including a $7.3\times$ speedup over other accelerated samplers in Magnetic Particle Imaging.