arxiv: 2605.03548 · v1 · submitted 2026-05-05 · 💻 cs.LG · cs.AI

Recognition: unknown

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

Han Wan, Hao Sun, Hao Zhou, Rui Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:56 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords rectified flowphysics-embedded modelsspatiotemporal reconstructionPDE reconstructionuncertainty quantificationsparse measurementsgenerative modelingconstraint projection

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The pith

PerFlow decouples observation conditioning from physics enforcement in rectified flows to enable fast sparse PDE reconstruction with invariance guarantees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to solve the slow and unstable inference in generative models for reconstructing PDE-governed spatiotemporal fields from sparse irregular measurements. Existing methods enforce data consistency and physics constraints together through repeated gradient guidance during sampling. PerFlow instead feeds observations directly into the rectified flow dynamics for guidance-free conditioning and applies a separate constraint-preserving projection to embed hard physics such as conservation or incompressibility. Theoretical invariance guarantees ensure that sampling trajectories remain on the physics-consistent manifold at every step. This separation yields competitive reconstruction accuracy, sound physical consistency, and inference that is up to 320 times faster than guided diffusion baselines using only 50 steps.

Core claim

PerFlow decouples observation conditioning from physics enforcement by incorporating measurements into the rectified-flow dynamics without guidance and embedding hard PDE constraints via a constraint-preserving projection operator. Theoretical analysis establishes invariance guarantees that keep trajectories on the physics-consistent manifold throughout sampling. Experiments across PDE systems confirm that the approach delivers competitive accuracy and physical consistency while enabling efficient conditional sampling and uncertainty quantification.

What carries the argument

The constraint-preserving projection operator embedded in the rectified flow sampling process, which enforces PDE constraints independently of the observation conditioning step.

If this is right

Reconstruction accuracy stays competitive with guided methods while physical consistency is maintained by design.
Uncertainty quantification becomes feasible because multiple conditional samples can be drawn in far fewer steps.
The method applies to various PDE systems with sparse and irregular observations without requiring per-sample optimization.
Inference speed improves dramatically, reaching 320 times faster than 2000-step guided diffusion baselines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The decoupling of conditioning and projection could be tested on other flow-based generative models to reduce guidance overhead in constrained generation tasks.
If efficient projections can be derived for additional PDE classes, the framework might support real-time physics-consistent forecasting from limited sensor data.
Checking preservation of invariants after exactly 50 steps on unseen PDEs would provide a direct test of whether the finite-step guarantees transfer.

Load-bearing premise

An efficient constraint-preserving projection operator exists for the target PDEs that can be applied without distorting the learned distribution and that the invariance guarantees hold during finite-step sampling.

What would settle it

Generated samples after 50 steps that violate a hard constraint such as mass conservation or incompressibility on a tested PDE system would show that the projection and invariance guarantees fail to maintain physics consistency in practice.

Figures

Figures reproduced from arXiv: 2605.03548 by Han Wan, Hao Sun, Hao Zhou, Rui Zhang.

**Figure 1.** Figure 1: (a) Relative ℓ2 error vs. sampling steps on Darcy (S3GM, DiffPDE, PerFlow). (b) DiffPDE sensitivity on Poisson: Relative ℓ2 error vs. PDE-loss weight λ. et al., 2020; Lu et al., 2021; Li et al., 2024b]. Most existing surrogate models are deterministic and trained on gridded inputs (e.g., complete initial conditions), yet real-world measurement data is often sparse and irregular, rendering field reconstru… view at source ↗

**Figure 2.** Figure 2: Overview of PerFlow. (a) PerFlow learns a rectified-flow transport between initial and target distributions on the physics-consistent manifold. (b) Training and inference pipelines. During training, sparse observations are formed by masking ground-truth fields, while priorprojected noise is combined with the target field using weights (1 − t) and t to construct xt. PerFlow learns a conditional velocity fi… view at source ↗

**Figure 3.** Figure 3: Sparse reconstruction physical fields (left) and the distribution of the point-wise mean view at source ↗

**Figure 4.** Figure 4: Uncertainty quantification under sparse observations, including Darcy coefficient view at source ↗

**Figure 5.** Figure 5: Reconstruction error versus sampling steps on ( view at source ↗

**Figure 6.** Figure 6: Poisson data-scaling study for PerFlow: test Rel- view at source ↗

**Figure 7.** Figure 7: Sparse reconstruction on four PDE benchmarks: view at source ↗

**Figure 8.** Figure 8: Darcy flow reconstruction under varying numbers of observations ( view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PerFlow inserts a projection step into rectified flows to hard-enforce physics constraints during sampling, which could cut inference time dramatically if the invariance holds in discrete steps.

read the letter

The paper's main contribution is a way to decouple observation conditioning from physics enforcement inside rectified flow. Instead of using gradient guidance at every step to satisfy both data and PDE constraints, they feed observations directly into the flow dynamics and apply a constraint-preserving projection (for things like incompressibility) after each update. They also give an invariance proof meant to keep the trajectory on the physics manifold throughout sampling. That setup is what lets them claim 50-step conditional sampling and speedups up to 320x over 2000-step guided diffusion baselines while still reporting competitive reconstruction accuracy on PDE examples.

Referee Report

2 major / 2 minor

Summary. The paper proposes PerFlow, a physics-embedded rectified flow model for reconstructing and quantifying uncertainty in spatiotemporal PDE-governed fields from sparse, irregular observations. It decouples observation conditioning (via guidance-free integration of observations into the rectified-flow dynamics) from physics enforcement (via insertion of a constraint-preserving projection operator, e.g., for incompressibility or conservation), while providing theoretical invariance guarantees that sampled trajectories remain on the physics-consistent manifold. Experiments across multiple PDE systems report competitive reconstruction accuracy, maintained physics consistency, efficient 50-step conditional sampling, and up to 320x faster inference relative to 2000-step guided diffusion baselines.

Significance. If the invariance guarantees and non-distorting projection hold under the finite-step discretizations used in practice, PerFlow would represent a meaningful advance in physics-informed generative modeling for inverse problems. By avoiding sampling-time gradient guidance, the method addresses documented issues of instability and computational cost in prior guided-diffusion approaches while enforcing hard constraints exactly (rather than softly). The combination of a theoretical invariance result with empirical demonstrations on diverse PDE systems and explicit speed/accuracy comparisons strengthens its potential impact in applications such as fluid dynamics and spatiotemporal forecasting.

major comments (2)

[Theoretical Analysis] Theoretical section establishing invariance guarantees: the proof is stated for the continuous-time rectified-flow ODE with an exact constraint-preserving projection. The manuscript does not supply a discrete-time error analysis or bound on manifold drift for the 50-step Euler (or similar) discretization employed in all experiments; without this, the central claim that trajectories 'remain exactly on the physics-consistent manifold' is not yet load-bearing for the finite-step regime that delivers the reported speedups.
[Method] Method section describing the projection operator and its insertion into the rectified-flow ODE: the assertion that the projection 'embeds hard physics without distorting the learned conditional distribution' requires explicit verification that the operator commutes with the learned velocity field at each discrete step. If the projection is only approximately divergence-free or mass-conserving (as is common on irregular grids), the decoupling between guidance-free conditioning and hard constraints may not be preserved, undermining the claimed advantage over guided baselines.

minor comments (2)

Abstract and experiments: the speedup is reported as '320 faster' (presumably 320x); clarify the exact metric (wall-clock time, FLOPs, or steps) and include error bars or multiple random seeds for all quantitative tables comparing reconstruction error and physics residuals.
Notation: the projection operator is introduced without a compact symbol or pseudocode; adding an explicit definition (e.g., P(·)) and a short algorithm box would improve readability when the operator is referenced in both the theory and sampling procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We have carefully considered the major comments and provide detailed responses below, along with plans for revisions to address the concerns raised.

read point-by-point responses

Referee: [Theoretical Analysis] Theoretical section establishing invariance guarantees: the proof is stated for the continuous-time rectified-flow ODE with an exact constraint-preserving projection. The manuscript does not supply a discrete-time error analysis or bound on manifold drift for the 50-step Euler (or similar) discretization employed in all experiments; without this, the central claim that trajectories 'remain exactly on the physics-consistent manifold' is not yet load-bearing for the finite-step regime that delivers the reported speedups.

Authors: We agree with the referee that extending the theoretical analysis to the discrete-time setting is important for rigorously supporting the claims in the practical finite-step regime. In the revised manuscript, we will add a discrete-time error analysis that bounds the manifold drift for the Euler discretization used in our 50-step sampling. This analysis will rely on the Lipschitz properties of the velocity field and the projection operator. Furthermore, we will augment the experimental section with plots and metrics showing the evolution of constraint violations (such as divergence or conservation errors) over the sampling trajectory, demonstrating that the drift remains negligible in practice. These changes will strengthen the connection between the continuous guarantees and the observed efficiency and accuracy. revision: yes
Referee: [Method] Method section describing the projection operator and its insertion into the rectified-flow ODE: the assertion that the projection 'embeds hard physics without distorting the learned conditional distribution' requires explicit verification that the operator commutes with the learned velocity field at each discrete step. If the projection is only approximately divergence-free or mass-conserving (as is common on irregular grids), the decoupling between guidance-free conditioning and hard constraints may not be preserved, undermining the claimed advantage over guided baselines.

Authors: We appreciate this insightful comment on the methodological details. The projection operator is designed as an exact retraction onto the physics manifold (e.g., via orthogonal projection for linear constraints like divergence-free fields), ensuring it does not distort the distribution in the continuous sense. To provide the requested verification, we will include in the revised method section a proposition establishing that the projection commutes with the velocity field updates in a manner that preserves the learned conditional distribution. For irregular grids where approximations are necessary, we will discuss the discretization errors and include new ablation experiments that quantify any impact on the decoupling and compare against guided baselines. This will clarify and bolster the advantages of our approach. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation builds independently on rectified flows and projections

full rationale

The paper claims a decoupling of conditioning and physics enforcement via guidance-free rectified-flow dynamics plus constraint-preserving projection, supported by invariance guarantees. No load-bearing step reduces by construction to fitted inputs, self-definitions, or self-citation chains; the theoretical guarantees and projection operator are presented as independent additions to existing rectified-flow frameworks. The approach remains self-contained against external benchmarks without renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from generative modeling and physics-informed methods plus the novel projection operator; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption A constraint-preserving projection operator exists and can be computed efficiently for the PDE systems of interest.
Invoked to embed hard physics and to establish invariance guarantees.
domain assumption Rectified flow dynamics remain stable when observations are injected directly without additional guidance.
Core premise enabling the decoupling of conditioning from physics enforcement.

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Case Batch size Num

Table 3: Training hyperparameters for different cases. Case Batch size Num. epochs Learning rate 1D Burgers 24 3001×10 −4 2D Darcy 24 5001×10 −4 2D Poisson 24 5001×10 −4 2D NS 10 8001×10 −4 We optimize all models using AdamW with weight decay1×10 −4. We adopt a learning-rate schedule with 10-epoch warmup followed by cosine decay from1×10 −4 to6×10 −5. We ...

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