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.

翻译：流体流动的统计代理建模具有挑战性，因为其动力学具有多尺度特性且对初始条件高度敏感。条件扩散代理模型可以做到精确，但通常需要数百次随机采样步骤。我们提出了一种整流流代理模型，它学习一个时间相关的条件速度场，该场沿着近似直线的轨迹传输输入到输出的分布规律。因此，推断过程是一个确定性的常微分方程求解，使得每次函数评估的信息量更大：在多尺度二维基准测试中，我们仅用 (8) 步常微分方程求解就达到了与扩散类模型相当的后验统计量，而基于分数的扩散模型则需要 (\ge 128) 步。理论上，我们为条件偏微分方程预测提供了分布层面的分析。我们 (i) 将单点 Wasserstein 场度量与统计解中的 (k=1) 相关-边缘视角联系起来，(ii) 推导出单步误差分解为一个**覆盖度**项（高频尾部，由结构函数/谱衰减控制）和一个**拟合**项（由训练目标控制），以及 (iii) 证明整流时间**直线度**控制着常微分方程的局部截断误差，从而为实际步长/步数选择提供指导。受此启发，我们引入了一种曲率感知采样器，它使用指数移动平均直线度代理在推断时自适应地调整混合比例和步长。在不可压缩和可压缩的多尺度二维流动中，该采样器在 Wasserstein 统计量和谱方面与扩散基线模型相当，在精细尺度结构保持上超越了均方误差代理模型，并显著降低了推断成本。