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

2025-12-10

AI Summary

Diffusion models suffer from spectral collapse in turbulent flow generation—where high-frequency structures are erroneously treated as noise and the inertial subrange’s −5/3 power-law spectrum is distorted—due to conventional Gaussian noise scheduling. To address this, we propose a physically grounded power-law noise schedule, β(τ) ∝ τ^γ, coupled with Lazy Diffusion, a single-step distillation technique. For the first time, our noise schedule is explicitly formulated as a Fourier-space, physics-driven spectral regularizer within a score-based diffusion framework. Leveraging SNR analysis in the frequency domain and geometric constraints on reverse trajectories, the method preserves multiscale energy cascade dynamics. Evaluated on 2D Kolmogorov turbulence and a 1/12° Gulf of Mexico ocean reanalysis task, our approach fully recovers the −5/3 power-law spectrum, enables stable autoregressive prediction over 100 steps, and reduces high-frequency energy error by 67%. This significantly enhances both physical fidelity and computational efficiency in multiscale turbulent flow modeling.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Planning, Routing, and Scheduling: Planning with Language Models
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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.

Problem

• Mitigating spectral collapse in diffusion models for turbulence
• Preserving fine-scale structure in generative flow emulation
• Enabling stable autoregression for multiscale dynamical systems

Innovation

• Power-law noise schedules preserve fine-scale structure
• Lazy Diffusion uses one-step distillation to bypass long trajectories
• Physics-aware diffusion processes restore realistic inertial-range scaling