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

arXiv: https://arxiv.org/abs/2609.40287 · HTML: https://mlsanity.com/paper/2609.40287/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 43 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 30 of 39)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)

## Abstract (arXiv)

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 Preconditioned Manifold one-step Flow Matching (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.