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

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.

💡 Research Summary

ProFlow introduces a principled zero‑shot sampling framework that enforces strict physical consistency and observational fidelity while preserving the statistical structure of a pre‑trained generative prior. The authors adopt Functional Flow Matching (FFM) as the prior: a neural network learns a continuous velocity field vθ(u, t) that transports a simple Gaussian random field (the reference measure μ0) to a target distribution μ1 supported on physically admissible fields. During training, FFM assumes a linear interpolation (“straight‑line bridge”) between reference and target samples, which yields a tractable probability path for sampling.

At inference time, ProFlow alternates two complementary steps at each discretized time tn:

1. Terminal proximal optimization – The current flow state ut is advanced by the learned velocity to obtain a candidate terminal state ŭ1 = ut + (1‑tn)vθ(ut, tn). Rather than accepting ŭ1 directly, the algorithm solves a proximal minimization problem: \