Flow-based generative models can face numerical challenges on scientific data with multiscale Fourier spectra, often producing large errors at fine scales. We approach this problem within the flow matching and stochastic interpolants framework, through the principled design of noise distributions and interpolation schedules. Working in function space ensures that the generative model remains well defined as the resolution is refined; the Lipschitz regularity of the drift is important to both this function-space well-posedness and the integration cost at fixed resolution. The central observation is that the noise should be at least as rough as the target distribution -- measured by Fourier-spectrum decay -- in order to keep the Lipschitz constant finite. For Gaussian and near-Gaussian targets whose fine-scale structure is known, matched-spectrum noise improves numerical efficiency over standard white-noise choices. For more complex non-Gaussian targets, matched-spectrum noise may not be sufficient, and we propose scale-adaptive interpolation schedules to mitigate the terminal-time stiffness that arises when the noise is rougher than the data. Numerical experiments on synthetic Gaussian random fields and on invariant measures of the stochastic Allen--Cahn and Navier--Stokes equations illustrate the approach and demonstrate its ability to generate high-fidelity samples at lower computational cost than traditional approaches.

翻译：基于流的生成模型在处理具有多尺度傅里叶谱的科学数据时可能面临数值挑战，通常在精细尺度上产生较大误差。我们在流匹配与随机插值框架内，通过噪声分布与插值调度的原理性设计来解决该问题。在函数空间中开展工作确保了生成模型在分辨率精化时仍然良定义；漂移项的Lipschitz正则性对函数空间良定性及固定分辨率下的积分成本均至关重要。核心发现是：为使Lipschitz常数保持有限，噪声应至少与目标分布具有相同粗糙度（以傅里叶谱衰减衡量）。对于精细尺度结构已知的高斯与近高斯目标，谱匹配噪声相比标准白噪声选择能提升数值效率。对于更复杂的非高斯目标，谱匹配噪声可能不足，我们提出尺度自适应插值调度以缓解噪声比数据更粗糙时产生的终端时间刚性。在合成高斯随机场及随机Allen–Cahn与Navier–Stokes方程的不变测度上的数值实验验证了该方法，并展示了其以低于传统方法的计算成本生成高保真样本的能力。