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

Physics-constrained flow matching seeks to preserve a target distribution while respecting nonlinear physical constraints. Extending physics-constrained generation from multi-step sampling to a one-step finite-interval map requires learning the full constrained transport in a single regression problem, which can become poorly conditioned due to residual geometry and anisotropic interpolation statistics. To address this issue, we introduce Preconditioned Manifold one-step Flow Matching (PMosFM), a preconditioned manifold matching framework for one-step physics-constrained generation. An exact residual chart parameterizes the feasible degrees of freedom, removing the corresponding residual-loss branch and terminal residual unrolling. A geometric preconditioner rescales feasible coordinates according to the geometry induced by physical decoding, while an interpolation-state preconditioner reduces anisotropy in the states presented to the network. Then a two-time objective combines velocity supervision with consistency between decoded physical endpoints. Our framework characterizes residual-normal curvature, separates geometric and statistical factors in local regression conditioning, and relates physical flow-map defects to endpoint distribution error. Matched ablations show that the precondition–manifold–matching construction reaches a fixed physical–distributional validation criterion with fewer optimizer updates and lower wall-clock training time. At inference, PMosFM replaces multi-step integration of the learned flow with one-step generation, requiring a single network evaluation followed by physical decoding. Experiments across representative PDE benchmarks show lower end-to-end sampling latency than the evaluated multi-step baselines at comparable physical and distributional fidelity.