DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

Huanshuo Dong; Hao Wu; Hong Wang; Qin-Yi Zhang; Zhezheng Hao

cs.LGMar 26, 2026

DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

Authors: Huanshuo Dong, Hao Wu, Hong Wang, Qin-Yi Zhang, Zhezheng Hao

Organizations: University of Science and Technology of China, Hefei, China · Tsinghua University, Beijing, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China · Zhejiang University, Hangzhou, China

Abstract

Long-term fluid dynamics forecasting is a critically important problem in science and engineering. While neural operators have emerged as a promising paradigm for modeling systems governed by partial differential equations (PDEs), they often struggle with long-term stability and precision. We identify two fundamental failure modes in existing architectures: (1) local detail blurring, where fine-scale structures such as vortex cores and sharp gradients are progressively smoothed, and (2) global trend deviation, where the overall motion trajectory drifts from the ground truth during extended rollouts. We argue that these failures arise because existing neural operators treat local and global information processing uniformly, despite their inherently different evolution characteristics in physical systems. To bridge this gap, we propose the Dual-Scale Neural Operator (DSO), which explicitly decouples information processing into two complementary modules: depthwise separable convolutions for fine-grained local feature extraction and an MLP-Mixer for long-range global aggregation. Through numerical experiments on vortex dynamics, we demonstrate that nearby perturbations primarily affect local vortex structure while distant perturbations influence global motion trends, providing empirical validation for our design choice. Extensive experiments on turbulent flow benchmarks show that DSO achieves state-of-the-art accuracy while maintaining robust long-term stability, reducing prediction error by over 88% compared to existing neural operators.

Figures & tables

Figure 2. Temporal evolution of vortex dipole under close (top) vs. far (bottom) perturbations. The curve represents a streamline of the flow field. The red-blue vortex pair on the left represents the dipole, while the isolated vortex on the right is the perturbing vortex.

Metric	Close ( $d=0.6$ )	Far ( $d=2.5$ )
Local gradient change $\Delta\max(\\|\nabla\omega\\|)$	+45%	$-$ 29%
Global position shift $\\|\Delta\mathbf{x}_{c}\\|$	0.8	0.6

Table 1. Comparison of effects for perturbations.

Figure 3. Overview of DSO architecture. The model consists of three components: (1) an encoder $\mathcal{E}$ that extracts multi-scale spatial features, (2) a translator $\mathcal{T}$ composed of stacked dual-pathway blocks, and (3) a decoder $\mathcal{D}$ that reconstructs the predicted field. Each block contains a Local Pathway (convolution) for fine-scale features and a Global Pathway (MLP-Mixer) for domain-wide information aggregation. The curved arrows connecting the encoder and decoder represent skip connections.

Method	NS-Forced				NS-Decaying
Method	All-step	1-step	10-step	19-step	All-step	1-step	50-step	99-step
DSO	0.0153	0.0001	0.0015	0.1086	0.1714	0.0003	0.1271	0.4986
UNO	0.2752	0.0037	0.0974	1.1894	nan	0.1771	nan	nan
CNO	0.3790	0.0008	0.1116	1.6521	4.5553	0.0390	5.2277	6.2792
FNO	0.0263	0.0001	0.0024	0.1852	1.6333	0.0242	1.7299	2.9465
U_Net	0.1278	0.0007	0.0308	0.6620	2.9288	0.0177	3.3029	4.6783

Table 2. MSE comparison across methods for NS-Forced and NS-Decaying turbulence. “All-step” denotes the average error over all prediction steps, while “

i

-step” indicates the error at the

i

-th specific prediction step. Lower MSE values indicate better performance, while "nan" indicates excessive model error or data overflow, suggesting model collapse.

Figure 4. We selected predictions from selected models on the NS-Decaying dataset, computed and analyzed the mean and variance of their SSIM values.

Method	All-step	1-step	50-step	99-step
DSO	0.1714	0.0003	0.1271	0.4986
w/o Conv	0.2059	0.0004	0.1565	0.5871
w/o MLP-Mixer	1.1335	0.0010	1.1330	2.4606

Table 3. Ablation study of DSO on the NS-Decaying dataset

Panel of Figure 5 — Figure 5. Comparison of gradient and divergence errors of different methods on the NS-Decaying dataset. (a) Gradient magnitude error quantifies the accuracy of capturing flow deformation features. (b) Divergence error measures physical consistency and mass conservation properties.

Figure 6. Prediction visualization of selected models across various time steps for the NS-Decaying case, where the top row displays the ground truth values.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Metric	Close ( $d=0.6$ )	Far ( $d=2.5$ )
Local gradient change $\Delta\\|\nabla\omega\\|_{\max}$	+45%	$-$ 29%
Global position shift $\\|\Delta\mathbf{x}_{c}\\|$	0.8	0.6

Appendix

Table 4. Comparison of local and global effects for close vs. far perturbations.

Model Name	Parameter	Value(s)
DSO	Input Shape	[1,1,64,64]
	Encode Hidden Dim	128
	Translator Hidden Dim	256
	Encode Layers	4
	Translator Layers	8
UNO	in_width	5

Appendix

Table 5. Model Parameters Configuration of NS-Forced

Model Name	Parameter	Value(s)
DSO	Input Shape	[1,1,128,128]
	Encode Hidden Dim	128
	Translator Hidden Dim	256
	Encode Layers	4
	Translator Layers	8
UNO	in_width	5

Appendix

Table 6. Model Parameters Configuration of NS-Decaying

Figure 7. Prediction visualization of selected models across various time steps for the NS-Forced case, where the top row displays the ground truth values.

Figure 8. Prediction visualization of selected models across various time steps for the NS-Decaying case, where the top row displays the ground truth values.

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Neural operators have emerged as powerful surrogates for dynamical systems due to their grid-invariant properties and computational efficiency. However, Fourier-based variants inherently truncate high-frequency components in spectral space, resulting in the loss of small-scale structures and degraded prediction quality at high resolutions when trained on low-resolution data. While diffusion-based enhancement methods can recover multi-scale features, they introduce substantial inference overhead that undermines the efficiency advantage of neural operators. In this work, we introduce MeanFlow-Enhanced Neural Operators (MENO), a novel framework that achieves accurate all-scale predictions with minimal inference cost. By leveraging the improved MeanFlow method, MENO restores both small-scale details and large-scale dynamics with superior physical fidelity and statistical accuracy. We evaluate MENO on three challenging dynamical systems, including phase-field dynamics, 2D Kolmogorov flow, and active matter dynamics, at resolutions up to 256

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Abstract

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