Rectified Flows for Fast Multiscale Fluid Flow Modeling

TL;DR

This paper introduces a rectified-flow surrogate model for multiscale fluid flow prediction that achieves high accuracy with fewer steps by learning a time-dependent velocity field, enabling deterministic inference and improved efficiency.

Contribution

The paper proposes a novel rectified-flow approach for fluid flow modeling that reduces inference steps and enhances accuracy over traditional diffusion-based methods.

Findings

Matches diffusion-class posterior statistics with only 8 ODE steps

Reduces inference cost significantly compared to diffusion methods

Preserves fine-scale structure beyond MSE surrogates

Abstract

Statistical surrogate modeling of fluid flows is hard because dynamics are multiscale and highly sensitive to initial conditions. Conditional diffusion surrogates can be accurate, but usually need hundreds of stochastic sampling steps. We propose a rectified-flow surrogate that learns a time-dependent conditional velocity field transporting input-to-output laws along near-straight trajectories. Inference is then a deterministic ODE solve, making each function evaluation more informative: on multiscale 2D benchmarks, we match diffusion-class posterior statistics with only (8) ODE steps versus (\ge 128) for score-based diffusion. Theoretically, we give a law-level analysis for conditional PDE forecasting. We (i) connect one-point Wasserstein field metrics to the (k=1) correlation-marginal perspective in statistical solutions, (ii) derive a one-step error split into a **coverage** term (high-frequency tail, controlled by structure functions/spectral decay) and a **fit** term (controlled by the training objective), and (iii) show that rectification-time **straightness** controls ODE local truncation error, yielding practical step-size/step-count guidance. Motivated by this, we introduce a curvature-aware sampler that uses an EMA straightness proxy to adapt blending and step sizes at inference. Across incompressible and compressible multiscale 2D flows, it matches diffusion baselines in Wasserstein statistics and spectra, preserves fine-scale structure beyond MSE surrogates, and significantly reduces inference cost.