One Scale at a Time: Scale-Autoregressive Modeling for Fluid Flow Distributions

TL;DR

The paper introduces scale-autoregressive modeling (SAR), a hierarchical approach for efficiently sampling unsteady fluid flow distributions on unstructured meshes, outperforming existing diffusion models in accuracy and speed.

Contribution

SAR is a novel coarse-to-fine hierarchical sampling method that reduces computational cost and improves accuracy for fluid flow distribution modeling.

Findings

SAR achieves lower distributional error than diffusion models.

SAR runs 2-7x faster than flow-matching Transolver.

SAR provides accurate statistical flow quantities in real-world benchmarks.

Abstract

Analyzing unsteady fluid flows often requires access to the full distribution of possible temporal states, yet conventional PDE solvers are computationally prohibitive and learned time-stepping surrogates quickly accumulate error over long rollouts. Generative models avoid compounding error by sampling states independently, but diffusion and flow-matching methods, while accurate, are limited by the cost of many evaluations over the entire mesh. We introduce scale-autoregressive modeling (SAR) for sampling flows on unstructured meshes hierarchically from coarse to fine: it first generates a low-resolution field, then refines it by progressively sampling higher resolutions conditioned on coarser predictions. This coarse-to-fine factorization improves efficiency by concentrating computation at coarser scales, where uncertainty is greatest, while requiring fewer steps at finer scales. Across unsteady-flow benchmarks of varying complexity, SAR attains substantially lower distributional error and higher per-sample accuracy than state-of-the-art diffusion models based on multi-scale GNNs, while matching or surpassing a flow-matching Transolver (a linear-time transformer) yet running 2-7x faster than this depending on the task. Overall, SAR provides a practical tool for fast and accurate estimation of statistical flow quantities (e.g., turbulent kinetic energy and two-point correlations) in real-world settings.