针对湍流模拟中扩散模型的频谱崩溃问题，提出懒惰扩散方法提升高阶波动保真度。
Lazy Diffusion: Mitigating spectral collapse in generative diffusion-based stable autoregressive emulation of turbulent flows
- 将噪声调度重解为频谱正则器，设计幂律调度以保留精细结构
- 在2D柯尔莫戈罗夫湍流与墨西哥湾海洋再分析中实现长时程自回归稳定
- 适合需要高保真多尺度动力系统建模的研究者使用
湍流具有宽带、幂律频谱特性，多尺度相互作用将高频扰动与大尺度动态耦合。尽管基于扩散的生成模型提供了严谨的概率预测框架，但标准DDPM会引发根本性‘频谱崩溃’：前向SDE的傅里叶分析显示，模式级信噪比随波数|k|单调衰减，当谱密度S(k)∝|k|⁻λ时，高频模式难以与噪声区分，导致内在频谱偏差。本文将噪声调度重新解释为频谱正则器，引入幂律调度β(τ)∝τ^γ，使细粒度结构在更长时间扩散中得以保留；并提出‘懒惰扩散’——一种单步蒸馏方法，利用学习到的得分几何绕过漫长逆时间轨迹，防止高波数退化。应用于高雷诺数二维柯尔莫戈罗夫湍流及1/12°墨西哥湾海洋再分析数据集，该方法有效缓解频谱崩溃，稳定长时程自回归过程，并恢复物理上合理的惯性区标度。结果表明，朴素高斯调度在结构上与幂律物理不兼容，而融入物理先验的扩散过程可生成准确、高效且完全概率化的多尺度动力系统替代模型。
原文摘要 · Abstract (English)
Turbulent flows posses broadband, power-law spectra in which multiscale interactions couple high-wavenumber fluctuations to large-scale dynamics. Although diffusion-based generative models offer a principled probabilistic forecasting framework, we show that standard DDPMs induce a fundamental \emph{spectral collapse}: a Fourier-space analysis of the forward SDE reveals a closed-form, mode-wise signal-to-noise ratio (SNR) that decays monotonically in wavenumber, $|k|$ for spectra $S(k)\!\propto\!|k|^{-λ}$, rendering high-wavenumber modes indistinguishable from noise and producing an intrinsic spectral bias. We reinterpret the noise schedule as a spectral regularizer and introduce power-law schedules $β(τ)\!\propto\!τ^γ$ that preserve fine-scale structure deeper into diffusion time, along with \emph{Lazy Diffusion}, a one-step distillation method that leverages the learned score geometry to bypass long reverse-time trajectories and prevent high-$k$ degradation. Applied to high-Reynolds-number 2D Kolmogorov turbulence and $1/12^\circ$ Gulf of Mexico ocean reanalysis, these methods resolve spectral collapse, stabilize long-horizon autoregression, and restore physically realistic inertial-range scaling. Together, they show that naïve Gaussian scheduling is structurally incompatible with power-law physics and that physics-aware diffusion processes can yield accurate, efficient, and fully probabilistic surrogates for multiscale dynamical systems.
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