PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation

Zhangyong Liang, Haibin Ling

Physics-constrained generative models aim to generate physical fields that match a target distribution and satisfy prescribed constraints. However, enforcing these constraints often increases sampling costs through iterative corrections or training costs through residual optimization and trajectory unrolling. To address this issue, we introduce \textbf{P}reconditioned \textbf{M}anifold \textbf{o}ne-\textbf{s}tep \textbf{F}low \textbf{M}atching (\textbf{PMosFM}), a preconditioned manifold matching framework for one-step physics-constrained generation. By encoding constraints in a manifold decoder, PMosFM learns transport in intrinsic coordinates without separate residual losses or terminal residual unrolling. A geometric preconditioner rescales coordinates using the decoder-induced metric, while a regularized covariance transform approximately whitens the interpolation-state inputs. A finite-interval objective couples velocity supervision with consistency between decoded endpoints in physical space. We show that exact parameterization removes residual-induced Gauss--Newton curvature, that geometric and covariance effects separate in a local conditioning bound, and that physical…