Flow Reasoning Models: Turning Flows Into Efficient Recurrent Reasoners

Structured reasoning requires making and revising interdependent decisions to reach a globally consistent solution. Existing architectures struggle with this: autoregressive models cannot revise emitted tokens, while masked diffusion models require careful decoding schemes to coordinate interdependent predictions. We introduce Flow Reasoning Models (FRMs), a novel framework for structured reasoning that adapts continuous flows over discrete structured outputs with a simple recurrent refinement mechanism. By self-conditioning a flow model on its own past outputs, we turn one-shot denoising into iterative solution refinement. This lets FRMs make and revise decisions in parallel, so interdependent decisions are updated jointly. Yet conventional self-conditioning becomes unreliable at greater recurrent depth due to exposure bias between one-step training predictions and recursively generated inference states. We address this mismatch with Fixed-Point Forcing (FPF), which trains FRMs on states produced by their own inference dynamics while preserving the standard flow-matching objective. FRMs achieve solve rates of $99.9\%$, $99.98\%$, and $100.0\%$ on Sudoku-Extreme, Zebra, and Maze-Unique, respectively. On Sudoku-Extreme, FRMs achieve higher solve rates than the evaluated masked-diffusion and specialized reasoning baselines while remaining highly compute-efficient, matching EqR's $99.79\%$ full-test solve rate with approximately $51\times$ fewer inference FLOPs.