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

Mario Lino; Nils Thuerey

cs.CEApr 13, 2026

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

Authors: Mario Lino, Nils Thuerey

Organizations: Technical University of Munich

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.

Figures & tables

Figure 1: (a) SAR generates resolution scales autoregressively from coarser to finer. (b) A SAR model is consists of a condition encoder, an autoregressive module, and a flow-matching sampler.

Panel of Table 1 — Table 1: Wasserstein-2 distance ( $W_{2}$ ) on the EllipseFlow datasets.

Figure 2: (a) Speed/distributional-accuracy trade-off on EllipseFlow-InDist , EllipseFlow-HighRe , and EllipseFlow-AoA10 . Curves for LDGN and LFM-GNN are obtained using 3, 5, 10, and 25 denoising steps. FMT curves use 3, 5, 10, 15, and 20 steps. The yellow SAR curve corresponds to using 2, 3, 5, and 10 denoising steps across all scales. The red SAR curve uses a different number of steps for each of the three scales: [2, 1, 1], [3, 2, 1], [5, 3, 1], and [10, 6, 1]. Inference times are measured on an NVIDIA RTX 3080. (b) Coefficient of determination ( $R^{2}$ ) for Turbulent Kinetic Energy (TKE) and Reynolds Shear Stress (SST) on the EllipseFlow-InDist dataset.

Figure 3: Speed and distributional/sample-accuracy trade-off on the Wing-TrainFullDist (training simulations extended to represent the full flow statistics) and Wing-InDist (design-space interpolation) datasets. Samples for DGN and LDGN are obtained using 5 denoising steps, and for LFM-GNN and FMT models using 3 steps. The yellow SAR values correspond to using 3 steps across all scales, while the red values correspond to 3, 2, and 1 steps in increasing resolution order. Inference times are measured on an NVIDIA RTX 3080. In this task, performance saturates quickly with the number of denoising steps. Because FMT models require very few steps, SAR’s computational advantage is reduced; however, SAR still achieves superior accuracy.

Figure 4: (a) Performance comparison between the latent SAR and non-latent SAR models across different numbers of denoising steps on the Ellipse-InDist dataset. (b) Probability density function comparison for a sample from the Ellipse-InDist dataset.

Figure 5: (a) Impact of ablation variants compared to our default configuration (striped bars), measured using Wasserstein-2 distance (top) and coefficient of determination (bottom). Bars report mean performance across the full test distributions from datasets EllipseFlow-InDist , EllipseFlow-HighRe , and EllipseFlow-AoA10 . (b) Visual comparison of SAR with (default) and without the condition encoder, shown on a representative sample from dataset EllipseFlow-AoA10 .

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Horizontal component of the velocity field at each of the three resolution scales ( $K=3$ ) for a sample from the EllipseFlow-InDist dataset.

Figure 7: Diffusion GNN ( Lino et al., 2025 ) and our SAR model both partition the node set

{\mathcal{V}}

into resolution scales but differ in their processing approach. Diffusion GNNs apply the same number of denoising steps across all scales using local message passing and local unpooling. In contrast, SAR allows fewer denoising steps at finer scales, making it feasible to use otherwise expensive attention. Upsampling is performed once per scale via the transformer based autoregressive module .

System	Node Condition Features ( $\bm{v}_{c,i}$ )	Outputs ( $\bm{s}_{i}$ )
Ellipse	Reynolds number ( $Re$ ); distances to top and bottom walls	Surface pressure ( $p_{i}$ )
EllipseFlow	Reynolds number ( $Re$ ); one-hot encoding of node type (inlet, ellipse boundary, interior)	Velocity components ( $u_{i}$ , $v_{i}$ ); pressure ( $p_{i}$ )
Wing	Outward unit normal vector of the wing surface	Surface pressure ( $p_{i}$ )

Appendix

Table 2: Conditioning node features and predicted outputs for each benchmark system.

Task	Mode	#Params	$F_{\text{model}}$	$F_{\text{emb}}$	$F_{\text{VAE}}$	$F_{L}$	$L_{\text{cond}}$	$L_{\text{AR}}$	$L_{\text{sampler}}$	#Scales
Ellipse	DGN	3.51 M	128	512	–	–	–	–	–	4
	LDGN	3.52 M	128	512	126	1	–	–	–	2 + 2
	FM-GNN	3.51 M	128	512	–	–	–	–	–	4
	LFM-GNN	3.52 M	128	512	126	1	–	–	–	2 + 2
	FMT-4	1.55 M	128	128	128	1	–	–	4	1
	SAR	1.83 M	128	128	128	1	2	2	2	3

Appendix

Table 3: Model size and hyperparameters.

Panel of Table 4 — Table 4: Wasserstein-2 distance ( $W_{2}$ ) on the Ellipse datasets.

Panel of Table 5 — Table 5: Coefficient of determination ( $R^{2}$ ) on the Ellipse datasets.

Panel of Table 6 — Table 6: Coefficient of determination ( $R^{2}$ ) on the EllipseFlow datasets.

Figure 8: (a) Impact of the number of scales, measured using the Wasserstein-2 distance. Bars indicate mean performance across the full test distributions from the Ellipse-InDist dataset. (b) Visual comparison of pressure profile samples predicted by SAR and LDGN ( Lino et al., 2025 ) for an ellipse from Ellipse-InDist with a relative thickness of 0.56 and

Re=736

Figure 9: Samples from LDGN ( Lino et al., 2025 ) , FMT-8, and SAR for (a) a simulation from the EllipseFlow-HighRe dataset, and (b) a simulation from the EllipseFlow-AoA10 dataset. SAR produces the most accurate samples across both settings.

Figure 10: Turbulent kinetic energy (top row), Reynolds shear stress (middle row), and inference time (bottom row) for the distributions predicted by LDGN, FMT-8, and SAR for a test case from the EllipseFlow-InDist dataset.

Figure 11: Speed/sample-accuracy trade-off on the EllipseFlow-InDist , EllipseFlow-HighRe , and EllipseFlow-AoA10 datasets. Curves for LDGN and LFM-GNN are obtained using 3, 5, 10, and 25 denoising steps. FMT curves use 3, 5, 10, 15, and 20 steps. The yellow SAR curve corresponds to using 2, 3, 5, and 10 denoising steps across all scales. The red SAR curve uses a different number of steps for each of the three scales: [2, 1, 1], [3, 2, 1], [5, 3, 1], and [10, 6, 1]. Inference times are measured on an NVIDIA RTX 3080.

Figure 12: (a) Standard deviation of pressure on a wing geometry unseen during training ( Wing-InDist dataset) from four sources: the ground-truth temporal distribution, the truncated training distribution, a flow-matching Transolver (FMT) model, and SAR. (b) For the same geometry, standard-deviation accuracy versus compute trade-off for the FMT and SAR models. FMT curves correspond to 3, 5, 10, 15, and 20 denoising steps. SAR curves use adaptive step configurations: [3, 2, 1], [5, 3, 1], [10, 3, 1], [15, 8, 2], and [20, 11, 2]. Inference times were measured on an NVIDIA RTX 3080.

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Spatiotemporal flows govern diverse phenomena across physics, biology, and engineering, yet modelling their multiscale dynamics remains a central challenge. Despite major advances in physics-informed machine learning, existing approaches struggle to simultaneously maintain long-term temporal evolution and resolve fine-scale structure across chaotic, turbulent, and physiological regimes. Here, we introduce Uni-Flow, a unified autoregressive-diffusion framework that explicitly separates temporal evolution from spatial refinement for modelling complex dynamical systems. The autoregressive component learns low-resolution latent dynamics that preserve large-scale structure and ensure stable long-horizon rollouts, while the diffusion component reconstructs high-resolution physical fields, recovering fine-scale features in a small number of denoising steps. We validate Uni-Flow across canonical benchmarks, including two-dimensional Kolmogorov flow, three-dimensional turbulent channel inflow generation with a quantum-informed autoregressive prior, and patient-specific simulations of aortic coarctation derived from high-fidelity lattice Boltzmann hemodynamic solvers. In the cardiovascular setting, Uni-Flow enables task-level faster than real-time inference of pulsatile hemodynamics, reconstructing high-resolution pressure fields over physiologically relevant time horizons in seconds rather than hours. By transforming high-fidelity hemodynamic simulation from an offline, HPC-bound process into a deployable surrogate, Uni-Flow establishes a pathway to faster-than-real-time modelling of complex multiscale flows, with broad implications for scientific machine learning in flow physics.

Abstract

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