MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systems

2026-04-08

AI Summary

This work addresses the challenge that existing Fourier-based neural operators struggle to preserve fine-scale structures at high resolution when trained on low-resolution data, while diffusion-based enhancement methods, though capable of recovering multiscale features, incur substantial inference overhead. To overcome this trade-off, the authors propose the MENO framework, which for the first time integrates a lightweight, modified MeanFlow mechanism into neural operators. This approach simultaneously models large-scale dynamics and reconstructs small-scale details with negligible additional inference cost. Experiments on phase-field systems, two-dimensional Kolmogorov flow, and active matter demonstrate that MENO achieves up to a twofold improvement in power spectral density accuracy and is 12 times faster at inference than DDIM, effectively balancing physical fidelity and statistical accuracy.

Abstract

Neural operators have emerged as powerful surrogates for dynamical systems due to their grid-invariant properties and computational efficiency. However, the Fourier-based neural operator framework inherently truncates 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 \textbf{M}eanFlow-\textbf{E}nhanced \textbf{N}eural \textbf{O}perators (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$256. Across all benchmarks, MENO improves the power spectrum density accuracy by up to a factor of 2 compared to baseline neural operators while achieving 12$\times$ faster inference than the state-of-the-art Diffusion Denoising 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.

Problem

• neural operators
• high-frequency truncation
• multi-scale prediction
• inference efficiency
• dynamical systems

Innovation

• Neural Operators
• MeanFlow
• Multi-scale Modeling
• Dynamical Systems
• Efficient Inference