PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

2026-05-05

AI Summary

Reconstructing spatiotemporal fields governed by partial differential equations (PDEs) from sparse and irregular observations is a highly ill-posed problem, wherein existing methods struggle to simultaneously achieve computational efficiency, physical consistency, and uncertainty quantification. This work proposes PerFlow—a physics-informed rectified flow model that decouples observational conditioning from physical laws and introduces an unguided conditional injection mechanism together with constraint-preserving projections to hard-embed priors such as incompressibility or conservation laws. By preserving the invariance of the physical manifold, PerFlow substantially enhances sampling efficiency and stability. Experiments demonstrate that PerFlow achieves high-fidelity, physically consistent reconstructions across diverse PDE systems with only 50 sampling steps, yielding up to a 320× speedup over a 2000-step guided diffusion baseline.

Abstract

Reconstructing PDE-governed fields from sparse and irregular measurements is challenging due to their ill-posed nature. Deterministic surrogates are trained on dense fields that struggle with limited measurements and uncertainty quantification. Generative models, by learning distributions over spatiotemporal fields, can better handle sparsity and uncertainty. However, existing generative approaches enforce data consistency and PDE constraints simultaneously via sampling-time gradient guidance, resulting in slow and unstable inference. To this end, we propose PerFlow, a Physics-embedded rectified Flow for efficient sparse reconstruction and uncertainty quantification of spatiotemporal dynamics. PerFlow decouples observation conditioning from physics enforcement, performing guidance-free conditioning by feeding observations into rectified-flow dynamics while embedding hard physics via a constraint-preserving projection (e.g., incompressibility or conservation). Theoretically, we establish invariance guarantees to ensure that trajectories remain on the physics-consistent manifold throughout sampling. Experiments on various PDE systems demonstrate competitive reconstruction accuracy with sound physics consistency, while enabling efficient conditional sampling (e.g., 50 steps) and up to 320 faster inference than 2000-step guided diffusion baselines.

Problem

• spatiotemporal dynamics
• PDE-governed fields
• sparse measurements
• uncertainty quantification
• ill-posed reconstruction

Innovation

• Physics-Embedded
• Rectified Flow
• Uncertainty Quantification
• Spatiotemporal Dynamics
• Constraint-Preserving Projection