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ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance

Zichao Yu, Ming Li, Wenyi Zhang, Difan Zou, Weiguo Gao

TL;DR

ProFlow is presented, a proximal guidance framework for zero-shot physics-consistent sampling, defined as inferring solutions from sparse observations using a fixed generative prior without task-specific retraining.

Abstract

Inferring physical fields from sparse observations while strictly satisfying partial differential equations (PDEs) is a fundamental challenge in computational physics. Recently, deep generative models offer powerful data-driven priors for such inverse problems, yet existing methods struggle to enforce hard physical constraints without costly retraining or disrupting the learned generative prior. Consequently, there is a critical need for a sampling mechanism that can reconcile strict physical consistency and observational fidelity with the statistical structure of the pre-trained prior. To this end, we present ProFlow, a proximal guidance framework for zero-shot physics-consistent sampling, defined as inferring solutions from sparse observations using a fixed generative prior without task-specific retraining. The algorithm employs a rigorous two-step scheme that alternates between: (\romannumeral1) a terminal optimization step, which projects the flow prediction onto the intersection of the physically and observationally consistent sets via proximal minimization; and (\romannumeral2) an interpolation step, which maps the refined state back to the generative trajectory to maintain consistency with the learned flow probability path. This procedure admits a Bayesian interpretation as a sequence of local maximum a posteriori (MAP) updates. Comprehensive benchmarks on Poisson, Helmholtz, Darcy, and viscous Burgers' equations demonstrate that ProFlow achieves superior physical and observational consistency, as well as more accurate distributional statistics, compared to state-of-the-art diffusion- and flow-based baselines.

ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance

TL;DR

ProFlow is presented, a proximal guidance framework for zero-shot physics-consistent sampling, defined as inferring solutions from sparse observations using a fixed generative prior without task-specific retraining.

Abstract

Inferring physical fields from sparse observations while strictly satisfying partial differential equations (PDEs) is a fundamental challenge in computational physics. Recently, deep generative models offer powerful data-driven priors for such inverse problems, yet existing methods struggle to enforce hard physical constraints without costly retraining or disrupting the learned generative prior. Consequently, there is a critical need for a sampling mechanism that can reconcile strict physical consistency and observational fidelity with the statistical structure of the pre-trained prior. To this end, we present ProFlow, a proximal guidance framework for zero-shot physics-consistent sampling, defined as inferring solutions from sparse observations using a fixed generative prior without task-specific retraining. The algorithm employs a rigorous two-step scheme that alternates between: (\romannumeral1) a terminal optimization step, which projects the flow prediction onto the intersection of the physically and observationally consistent sets via proximal minimization; and (\romannumeral2) an interpolation step, which maps the refined state back to the generative trajectory to maintain consistency with the learned flow probability path. This procedure admits a Bayesian interpretation as a sequence of local maximum a posteriori (MAP) updates. Comprehensive benchmarks on Poisson, Helmholtz, Darcy, and viscous Burgers' equations demonstrate that ProFlow achieves superior physical and observational consistency, as well as more accurate distributional statistics, compared to state-of-the-art diffusion- and flow-based baselines.
Paper Structure (37 sections, 1 theorem, 43 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 37 sections, 1 theorem, 43 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works and our contributions
  3. Classical numerical solvers
  4. Physics-informed deep learning
  5. Neural operator learning
  6. Generative modeling for PDEs
  7. Contributions of this work
  8. Functional flow matching prior for PDEs
  9. Method
  10. Problem statement
  11. Derivation of the proximal guidance scheme
  12. Terminal update as a local MAP step
  13. Interpolation step and compatibility with the flow
  14. Algorithmic framework of ProFlow
  15. Terminal optimization step
  16. ...and 22 more sections

Key Result

Proposition 4.1

Let $0 \leq t < s \leq 1$ and let $\mathcal{C}$ and $\mathcal{O}$ be the physical consistency event and observation event, respectively. Assume that under the conditional measure $\mathbb{P}(\cdot \mid \mathcal{C}, \mathcal{O})$ the joint law of $(\bm u_t, \bm u_s, \bm u_1)$ admits conditional densi

Figures (4)

  • Figure 1: A conceptual illustration of ProFlow. The algorithm alternates between (i) a terminal optimization step, which projects the model's predicted solution $\hat{\bm{u}}_1$ onto the physically and observationally consistent set by solving a proximal optimization problem; and (ii) an interpolation step, which constructs the next state $\bm{u}_{t_{n+1}}$ by linearly blending the refined solution $\bm{u}_1$ with the freshly drawn noise $\bm{\varepsilon}$, adhering to the flow matching probability path.
  • Figure 2: Visual comparison of the forward problem on the Darcy equation. The leftmost column displays the input coefficient field. The subsequent columns compare the Ground Truth solution against predictions from ProFlow and baseline approaches (ECI, DiffusionPDE, D-Flow, and PCFM). ProFlow predicts the solution field with high visual fidelity, capturing details that more closely match the ground truth than baseline methods.
  • Figure 3: Visual comparison of generated solutions for the Burgers' equation. The top section displays results for the Initial Condition (IC) configuration, while the bottom section displays the Boundary Condition (BC) configuration. The leftmost column visualizes the sparse input provided to the models. Since the provided IC/BC data does not uniquely constrain the PDE solution, the reconstruction is ill-posed. To capture this uncertainty, we generate four distinct samples for each method by varying the initial random noise. While baselines like PCFM exhibit mode collapse (producing identical samples) or artifacts, ProFlow generates diverse, physically plausible trajectories consistent with the governing PDE.
  • Figure 4: Visual comparison of sparse-in-time reconstruction on the Burgers' equation. The leftmost column visualizes the sparse input data, consisting of observations at only five randomly sampled timestamps. The subsequent columns contrast the ground truth spatiotemporal solution with reconstructions from ProFlow and baseline methods (ECI, DiffusionPDE, D-Flow, PCFM). ProFlow accurately interpolates the shock propagation dynamics between observed timestamps, whereas baselines exhibit significant blurring or spurious oscillations in the unobserved intervals.

Theorems & Definitions (4)

  • Remark 4.1: Extension to time-dependent problems
  • Proposition 4.1: Conditional law factorization
  • proof
  • Remark 4.2: Classification as a hard constraint method