Lazy Diffusion: Mitigating spectral collapse in generative diffusion-based stable autoregressive emulation of turbulent flows

TL;DR

This work identifies a fundamental spectral bias in diffusion-based turbulence surrogates, showing that standard Gaussian noise schedules induce a wavenumber-dependent spectral collapse that erases high-k information. It reframes the forward diffusion as a spectral regularizer and introduces power-law noise schedules to preserve fine-scale structure, complemented by Lazy Diffusion, a one-step distillation to bypass long reverse trajectories. The authors validate these ideas on high-Reynolds 2D Kolmogorov turbulence and Gulf of Mexico ocean reanalysis, demonstrating restored inertial-range scaling, stable long-horizon autoregression, and substantial computational gains. Collectively, the approach offers a physics-informed diffusion framework capable of accurate, efficient probabilistic surrogates for multiscale dynamical systems.

Abstract

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.

Figures (9)

• Figure 1: Schematic of the diffusion model and the lazy diffusion distillation.
• Figure 2: Signal to Noise Ratio for low, medium, and high k frequency bands for a power law adopting signal using different power values $\gamma$. SNR = 0 represents the white noise floor of the DDPM process.
• Figure 3: Autoregressive evolution of vorticity fields in System 1. Snapshots shown at $t = \{1, 100, 500, 1000, 2000\}$ timesteps ($t=20\Delta t_{DNS}$) for the ground truth, linear‐schedule DDPM, the power‐schedule model with $\gamma = 5.0$, and its lazy‐diffusion variant.
• Figure 4: Quantitative evaluation metrics over 2000-timestep autoregressive rollouts for linear ($\gamma = 1$ ) DDPM, $\gamma = 5.0$ power schedule, and Lazy retraining ($\gamma = 5.0$). Row 1 shows Relative RMSE at each timestep. Row 2 Shows the total histogram of of vorticity values over all autoregression times. Row 3 shows the latitudinal spectrum averaged over the rollout. Row 4 shows the turbulent kinetic energy (TKE) spectra averaged over the whole rollout.
• Figure 5: Power schedule analysis and spectral performance. (a) Relative spectral error as a function of power exponent $\gamma$, showing optimal performance at $\gamma = 5.0$ and catastrophic failure above $\gamma = 6.0$. Orange triangles indicate lazy diffusion retraining. (b) Signal-to-noise ratio evolution, $\alpha(\tau)$, for different power schedules, with higher $\gamma$ delaying noise injection. (c) Time-averaged latitudinal spectra of vorticity. (d) TKE spectra for all models demonstrate that appropriate power scheduling recovers high-wavenumber dynamics.