MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systems

Tianyue Yang; Xiao Xue

cs.LGApr 8, 2026

MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systems

Authors: Tianyue Yang, Xiao Xue

Organizations: Centre for Computational Science, University College London, London, United Kingdom

Abstract

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 $\times$ 256. Across all benchmarks, MENO improves the power spectrum density accuracy by up to a factor of 2 compared to baseline neural operators while achieving up to $14\times$ faster inference than the state-of-the-art Denoising Diffusion Implicit Model (DDIM)-enhanced counterparts, effectively bridging the gap between accuracy and efficiency. The flexibility and efficiency of MENO position it as an efficient surrogate model for scientific machine learning applications where both statistical integrity and computational efficiency are paramount.

Figures & tables

Figure 1: The MeanFlow-Enhanced Neural Operators framework. Panel (a) illustrates training the MeanFlow decoder on high-resolution fields by learning denoising trajectories. Panel (b) shows training an autoregressive neural operator on low-resolution data. Panel (c) combines both components into the full MENO pipeline: given a low-resolution initial condition, the neural operator produces a low-resolution rollout, which is then decoded into a high-resolution prediction by the MeanFlow decoder.

Figure 2: Relative $L_{2}$ loss over rollout time (top row) and the corresponding wavenumber energy spectra (bottom row). Shaded regions denote the standard error of the mean (SEM) computed over multiple test trajectories (PF100: 10, KF256: 8, AM256: 25). (a) PF100 $20\rightarrow 100$ : a FNO model trained on $20\times 20$ data and its enhanced variants. (b) KF256 $64\rightarrow 256$ : performance of a UNO model trained on $64\times 64$ data and its enhanced variants. (c) AM256 $64\rightarrow 256$ : performance of a UNO model trained on $64\times 64$ data and its enhanced variants.

	$20\rightarrow 100$			$50\rightarrow 100$
Model	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$
Neural operators without enhancement
FNO	0.64	0.60	$4.98\times 10^{-5}$	0.26	0.85	$1.99\times 10^{-5}$
Diffusion-enhanced neural operators
DM-FNO	0.50	0.65	$2.97\times 10^{-5}$	0.30	0.77	$2.28\times 10^{-5}$
DM-UNO	0.54	0.63	$3.34\times 10^{-5}$	0.32	0.76	$2.40\times 10^{-5}$

Table 1: PF100 prediction metrics for direct neural operator super-resolution, diffusion-enhanced neural operators, and MENO. The best value in each column is shown in bold. The stand-alone UNO super-resolution model was not included for PF100 because it did not produce reliable autoregressive rollouts under the same training setting as FNO. Relative improvement is computed with respect to the corresponding direct neural operator baseline.

Figure 3: Free-energy trajectories for all models and resolution settings on PF100. Upper panels: UNO-based models; lower panels: FNO-based models. Left column: $20\rightarrow 100$ ; right column: $50\rightarrow 100$ . Only the MENO variants consistently recover the correct free-energy evolution. Shaded regions denote the SEM over 10 test trajectories.

Figure 4: Temporal autocorrelation functions on PF100 across all resolution settings and model variants. Shaded bands denote the SEM over 10 test trajectories. MENO best preserves the long-time temporal coherence of the phase-field dynamics.

Figure 5: Snapshots along an uncurated PF100 trajectory for the $50\rightarrow 100$ setting. The top row presents the high-resolution ground truth, and the next three rows visualize absolute errors for NO-SR, MENO-NO, and DM-NO, respectively. MENO produces the lowest interface-localized error and best preserves the evolving phase morphology.

Model	Parameters, NO + decoder	Inference time per frame (s)
Diffusion-enhanced neural operators
DM-FNO	6.33M + 1.44M	$0.176\pm 0.022$
DM-UNO	6.38M + 1.44M	$0.178\pm 0.021$
MENO models
MENO-FNO	6.33M + 1.44M	$\mathbf{0.028\pm 0.022}$
MENO-UNO	6.38M + 1.44M	$0.031\pm 0.021$

Table 2: PF100 model size and inference time for generative-enhanced neural operators. Timings are measured per predicted frame for the full pipeline, including the neural operator and the decoder.

Figure 6: Snapshots along a trajectory from the KF256 dataset at $t=6,12,\ldots,54$ . The top row reports the high-resolution ground truth at $256\times 256$ . The next three rows visualize the absolute errors of UNO-SR, MENO-UNO, and DM-UNO, respectively. The color bars indicate the vorticity field and absolute error magnitude.

Figure 7: Temporal autocorrelation functions on KF256 across all resolution settings and model variants. Shaded bands denote the SEM over 8 test trajectories. The early-time decay is emphasized because it captures the dominant temporal memory of the turbulent vorticity field.

Neural operators without enhancement
	$32\rightarrow 256$			$64\rightarrow 256$
Model	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$
FNO	$\mathbf{0.19}$	0.73	$1.22\times 10^{-5}$	0.13	0.82	$1.00\times 10^{-5}$
UNO	0.23	0.66	$1.74\times 10^{-5}$	0.22	0.71	$1.76\times 10^{-5}$
Diffusion-enhanced neural operators
DM-FNO	0.34	0.64	$8.79\times 10^{-6}$	0.27	0.72	$6.70\times 10^{-6}$

Table 3: KF256 prediction metrics for direct neural operator super-resolution, diffusion-enhanced neural operators, and MENO.

\mathrm{R}L_{2}

and SSIM are computed over the first 20 frames, while PSDD is computed over the full 180-frame trajectories. The best value in each column is shown in bold. Relative improvement is computed with respect to the corresponding direct neural operator baseline.

Model	Parameters, NO + decoder	Inference time per frame (s)
Diffusion-enhanced neural operators
DM-FNO	27.10M + 6.63M	$0.377\pm 0.008$
DM-UNO	26.83M + 6.63M	$0.376\pm 0.008$
MENO models
MENO-FNO	27.10M + 6.63M	$\mathbf{0.030\pm 0.004}$
MENO-UNO	26.83M + 6.63M	$\mathbf{0.030\pm 0.004}$

Table 4: KF256 model size and inference time for generative-enhanced neural operators. Timings are measured per predicted frame for the full pipeline, including the neural operator and the decoder.

Figure 8: Snapshots along an uncurated AM256 trajectory for the $64\rightarrow 256$ setting. The top row presents the high-resolution ground truth, and the next three rows visualize absolute errors for UNO-SR, MENO-UNO, and DM-UNO, respectively. Frames are normalized per frame to highlight error evolution since the mean concentration drifts over time.

Figure 9: Temporal autocorrelation functions on AM256 across all resolution settings and model variants. Shaded bands denote the SEM over 25 test trajectories. The early-time decay is emphasized because it captures the dominant temporal coherence of the active concentration field.

Model	Parameters, NO + decoder	Inference time per frame (s)
Diffusion-enhanced neural operators
DM-UNO	26.83M + 6.63M	$0.456\pm 0.011$
MENO models
MENO-UNO	26.83M + 6.63M	$\mathbf{0.032\pm 0.005}$

Table 5: AM256 model size and inference time for generative-enhanced neural operators. Timings are measured per predicted frame for the full pipeline, including the neural operator and the decoder.

Neural operators without enhancement
	$32\rightarrow 256$			$64\rightarrow 256$
Model	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$
UNO	$2.11\times 10^{-3}$	0.72	$1.09\times 10^{-5}$	$2.05\times 10^{-3}$	0.73	$1.12\times 10^{-5}$
Diffusion-enhanced neural operators
DM-UNO	$2.17\times 10^{-3}$	0.56	$1.24\times 10^{-5}$	$2.14\times 10^{-3}$	0.56	$1.30\times 10^{-5}$
MENO models

Table 6: AM256 prediction metrics for direct neural operator super-resolution, diffusion-enhanced neural operators, and MENO.

\mathrm{R}L_{2}

and SSIM are computed over the first 10 frames, while PSDD is computed over the full 80-frame trajectories. The best value in each column is shown in bold. Relative improvement is computed with respect to the corresponding direct neural operator baseline.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 10: Image adapted from [ 32 ] , showing the reconstruction error trends for the $32\rightarrow 256$ generative refinement task. The legend compares three reconstruction strategies; in this work, we focus on the Baseline method.

Figure 11: Absolute $L_{2}$ loss of MENO decoder models on the KF256 dataset for the $32\rightarrow 256$ and $64\rightarrow 256$ generative refinement tasks.

Figure 12: Long range relative $L_{2}$ loss of all UNO-based models on the KF256 dataset for the $32\rightarrow 256$ generative refinement tasks. Shaded regions denote the SEM computed over 8 test trajectories.

i-MF variants
Model	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$
i-MF:MENO-FNO	$0.21$	$0.76$	$8.62\times 10^{-6}$
i-MF:MENO-UNO	$0.21$	$0.76$	$9.06\times 10^{-6}$
MF variants
MF:MENO-FNO	0.23	0.75	$8.96\times 10^{-6}$
MF:MENO-UNO	0.25	0.74	$9.21\times 10^{-6}$

Appendix

Table 7: Ablation study on the KF256 dataset comparing i-MF- and MF-based variants of MENO. All results are reported for the

32\rightarrow 256

setting.

Figure 13: Training loss curves of i-MF and the original MF on the KF256 dataset, shown with moving-average smoothing. The loss of the original MeanFlow increases as training proceeds, consistent with the behavior reported in Figure 3 of [ 13 ] .

Model	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$	$\mathrm{R}L_{2}\downarrow$	SSIM $\uparrow$	PSDD $\downarrow$
	32 $\rightarrow$ 256			64 $\rightarrow$ 256
DM-UNO	0.3541(4)	0.6383(2)	$(9.221\pm 0.001)\times 10^{-6}$	0.2581(1)	0.7178(3)	$(5.764\pm 0.001)\times 10^{-6}$
MENO-UNO	0.2131(1)	0.7612(1)	$(9.058\pm 0.001)\times 10^{-6}$	0.0813(1)	0.9321(2)	$(5.161\pm 0.001)\times 10^{-6}$

Appendix

Table 8: KF256

32\rightarrow 256

metrics for DM-enhanced NOs, and MENOs.

\mathrm{R}L_{2}

and SSIM are computed over the first 20 frames, while PSDD is computed over the full trajectories (180 frames). Uncertainties are computed over 100 runs with different random seeds. We report values to four significant figures; uncertainties smaller than this precision are rounded and reported as

\pm 1

in the last digit.

Parameter Name	FNO Configuration	UNO Configuration
channel_mlp_dropout	0	0
channel_mlp_expansion	0.5	0.5
channel_mlp_skip	soft-gating	linear
factorization	tucker	tucker
fno_skip	linear	linear
hidden_channels	78	128

Appendix

Table 9: FNO and UNO architectures used in all tasks for PF100 dataset.

UNet hyperparameters	Model setting hyperparameters
attention_resolutions: (16) channel_multipliers: (1, 1, 2) latent_dims: 32 num_res_blocks: 3 type: UNet	beta_end: 0.02 beta_start: 0.0001 diffusion_steps: 1000

Appendix

Table 10: Diffusion model setting for the PF100 experiment.

Parameter Name	FNO Configuration	UNO Configuration
channel_mlp_dropout	0	0
channel_mlp_expansion	2	2
channel_mlp_skip	soft-gating	linear
factorization	tucker	tucker
fno_skip	linear	linear
hidden_channels	84	64

Appendix

Table 11: FNO and UNO architectures used in all tasks for KF256 dataset.

UNet hyperparameters	Model setting hyperparameters
attention_resolutions: (16) channel_multipliers: (1, 1, 1, 2) latent_dims: 64 num_res_blocks: 3 type: UNet	beta_end: 0.02 beta_start: 0.0001 diffusion_steps: 1000

Appendix

Table 12: Diffusion model setting for the PF100 experiment.

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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.

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