# Scale-Adaptive Generative Flows for Multiscale Scientific Data

arXiv: https://arxiv.org/abs/2509.02971 · HTML: https://mlsanity.com/paper/2509.02971/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 65 references that are in the map.
- First posted on arXiv 3 Sep 2025.

## Builds on (most cited first, 30 of 45)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [Wavelet Diffusion Models are fast and scalable Image Generators](https://mlsanity.com/paper/2211.16152/) (2022)
- [Noise Estimation for Generative Diffusion Models](https://mlsanity.com/paper/2104.02600/) (2021)
- [Blurring Diffusion Models](https://mlsanity.com/paper/2209.05557/) (2022)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)

## Built on by (most cited first, 5 of 5)

- [Variational Optimality of F"ollmer Processes in Generative Diffusions](https://mlsanity.com/paper/2602.10989/) (2026)
- [Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference](https://mlsanity.com/paper/2604.20775/) (2026)
- [The critical slowing down in training diffusion models](https://mlsanity.com/paper/2605.12597/) (2026)
- [Discretization and Statistical Consistency of Functional Flow Matching](https://mlsanity.com/paper/2608.04531/) (2026)
- [Scale-Recursive Rectified Flows for Few-Step Precipitation Ensembles](https://mlsanity.com/paper/2610.02611/) (2026)

## Abstract (arXiv)

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.