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

arXiv: https://arxiv.org/abs/2512.09572 · HTML: https://mlsanity.com/paper/2512.09572/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 12 of 12)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution](https://mlsanity.com/paper/2304.07029/) (2023)
- [OceanNet: A principled neural operator-based digital twin for regional oceans](https://mlsanity.com/paper/2310.00813/) (2023)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos](https://mlsanity.com/paper/2503.18731/) (2025)
- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Long-term stability and generalization of observationally-constrained stochastic data-driven models for geophysical turbulence](https://mlsanity.com/paper/2205.04601/) (2022)

## Built on by (most cited first, 4 of 4)

- [Generative forecasting with joint probability models](https://mlsanity.com/paper/2512.24446/) (2025)
- [High-resolution probabilistic estimation of three-dimensional regional ocean dynamics from sparse surface observations](https://mlsanity.com/paper/2604.02850/) (2026)
- [Eigenanalysis framework for autoregressive neural emulators of multi-scale chaotic dynamics](https://mlsanity.com/paper/2608.16084/) (2026)
- [AI Emulation of Stochastic Sudden Stratospheric Warming with Interpretable Latent Structure](https://mlsanity.com/paper/2610.02069/) (2026)

## Abstract (arXiv)

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 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!τγ that preserve fine-scale structure deeper into diffusion time, along with 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 na"ive 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.