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

2026-09-30

AI Summary

Physics-constrained generative models often incur prohibitive sampling and training costs due to the enforcement of physical constraints. This work proposes PMosFM, a framework that achieves single-step physics-constrained generation via preconditioned manifold matching. Specifically, physical constraints are directly encoded into a manifold decoder, thereby eliminating residual losses and trajectory unrolling. Furthermore, geometric preconditioning and regularized covariance transformations are introduced to decouple geometric from covariance effects and optimize the condition number for transport learning. Experimental results demonstrate that PMosFM substantially reduces both training and sampling time as well as memory overhead, while maintaining equivalent physical and distributional fidelity compared to existing methods.

Abstract

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 flow-map error bounds endpoint distributional error. Controlled ablations examine conditioning, and experiments evaluate optimizer-update time and memory footprint. At inference, PMosFM uses one neural transport evaluation followed by physical decoding. Experiments across benchmarks show lower training and sampling time than the multi-step baselines at comparable physical and distributional fidelity. Code and datasets will be released publicly.

Problem

• Physics-constrained generation
• Flow matching
• Sampling cost
• Training cost
• Manifold

Innovation

• Manifold Matching
• Flow Matching
• Physics-Constrained Generation
• Geometric Preconditioner
• One-Step Generation