Unrolled Networks are Conditional Probability Flows in MRI Reconstruction

Magnetic Resonance Imaging (MRI) offers excellent soft-tissue contrast without ionizing radiation, but its long acquisition time limits clinical utility. Recent methods accelerate MRI by under-sampling $k$-space and reconstructing the resulting images using deep learning. Unrolled networks have been widely used for the reconstruction task due to their efficiency, but suffer from unstable evolving caused by freely-learnable parameters in intermediate steps. In contrast, diffusion models based on stochastic differential equations offer theoretical stability in both medical and natural image tasks but are computationally expensive. In this work, we introduce flow ODEs to MRI reconstruction by theoretically proving that unrolled networks are discrete implementations of conditional probability flow ODEs. This connection provides explicit formulations for parameters and clarifies how intermediate states should evolve. Building on this insight, we propose Flow-Aligned Training (FLAT), which derives unrolled parameters from the ODE discretization and aligns intermediate reconstructions with the ideal ODE trajectory to improve stability and convergence. Experiments on three MRI datasets show that FLAT achieves high-quality reconstructions with up to $3\times$ fewer iterations than diffusion-based generative models and significantly greater stability than unrolled networks.