MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systems
MENO combines improved MeanFlow with neural operators for high-resolution dynamical system prediction, achieving 2× spectral accuracy and 14× faster inference.
Key Findings
Methodology
The proposed MENO framework integrates a neural operator Gϕ trained to model coarse system dynamics with an improved MeanFlow decoder uθ for one-step high-resolution detail recovery. The neural operator learns the low-resolution flow map via autoregressive training, while the MeanFlow decoder, based on flow matching, models the average velocity over a time interval for efficient single-step generation. During inference, the low-res trajectory is generated by Gϕ, then refined into high-res fields with uθ, combining speed with detail preservation. Experiments on phase-field, 2D Kolmogorov flow, and active matter systems demonstrate spectral accuracy improvements up to 2× and inference speedups up to 14× over diffusion-based methods.
Key Results
- Across all benchmarks, MENO reduces spectral density error by over 50%, with SSIM reaching 0.86 and PSDD halved, at 256×256 resolution. Inference is 14 times faster than diffusion models, maintaining physical fidelity and capturing small-scale structures effectively.
- In phase-field simulations, free energy trajectories predicted by MENO closely match ground truth, with energy errors dropping from 0.64 to 0.21, and interface details sharply reconstructed. Spectral analysis shows broader energy distribution over wavenumbers, indicating better multi-scale fidelity.
- For turbulent and active matter systems, MENO outperforms baseline neural operators in statistical metrics, accurately reproducing long-term dynamics and spatial correlations, validating its robustness and generalization across physics domains.
Significance
This work addresses the longstanding challenge of balancing accuracy and efficiency in high-resolution dynamical system modeling. By combining neural operators with a fast, stable generative decoder, MENO overcomes spectral truncation limitations and inference bottlenecks of existing methods. Its ability to deliver physically consistent, detailed predictions at high resolution with minimal computational cost opens new avenues for real-time scientific simulations, climate modeling, and materials design. The framework’s flexibility suggests broad applicability across complex multi-scale systems, marking a significant step forward in scientific machine learning.
Technical Contribution
The key technical innovation lies in integrating an improved MeanFlow model into the neural operator pipeline, enabling single-step high-resolution detail recovery without iterative denoising. The two-stage architecture—coarse prediction via neural operator and detail refinement via flow matching—provides a unified, efficient solution. Theoretical guarantees on spectral fidelity and stability are supported by empirical results, demonstrating superior performance over diffusion-based generative models in both accuracy and inference speed. This approach opens new engineering possibilities for scalable, physics-informed generative models.
Novelty
This study is the first to embed an improved MeanFlow flow-matching based generator into neural operator frameworks for scientific prediction. Unlike traditional diffusion models requiring multiple denoising steps, MENO’s single-step refinement achieves comparable or better detail recovery with vastly reduced inference time. Its innovative combination of spectral neural operators and flow-based generative decoders sets a new standard for high-fidelity, efficient simulation of complex dynamical systems, bridging the gap between accuracy and computational practicality.
Limitations
- The current model’s performance in highly nonlinear, high-dimensional systems remains to be tested; its effectiveness may diminish when system dynamics deviate significantly from training data.
- Dependence on pre-trained decoders means that large system changes require retraining or fine-tuning, limiting adaptability in some scenarios.
- Handling multi-physics coupling with strong nonlinear interactions may introduce stability issues or detail loss, necessitating further robustness improvements.
Future Work
Future research will focus on extending MENO’s applicability to more complex, multi-physics systems, incorporating physics-informed constraints for improved physical consistency, and exploring unsupervised or semi-supervised training strategies to reduce data requirements. Additionally, integrating adaptive sampling and online learning could enhance robustness and generalization, making the framework suitable for real-time, large-scale scientific applications.
AI Executive Summary
High-fidelity simulation of complex dynamical systems remains a core challenge in computational science, especially when balancing accuracy with computational efficiency. Traditional numerical methods like DNS provide detailed results but are prohibitively expensive for large-scale or real-time applications. Neural operators, such as Fourier Neural Operator and DeepONet, have emerged as promising surrogates due to their resolution-invariance and fast inference, yet they suffer from spectral truncation, which leads to loss of small-scale features at high resolutions.
Diffusion-based generative models can recover multi-scale details effectively, but their iterative multi-step sampling process incurs significant computational costs, limiting real-time deployment. To address these issues, the authors propose MENO, a hybrid framework that combines a neural operator with an improved flow-matching-based MeanFlow decoder. This design decouples the coarse prediction and fine detail recovery, enabling single-step high-resolution generation with minimal overhead.
The core technical principle involves training a neural operator Gϕ to model low-resolution system dynamics, then employing the MeanFlow uθ to perform a one-step refinement that restores high-frequency details. This approach leverages the spectral fidelity of neural operators and the efficiency of flow matching, resulting in a method that is both accurate and fast. Extensive experiments on three benchmark systems demonstrate that MENO reduces spectral errors by over 50%, doubles the accuracy, and accelerates inference by up to 14 times compared to diffusion models.
This advancement has significant implications for scientific machine learning, enabling real-time, high-resolution simulations across physics, climate, and materials science. The framework’s flexibility allows adaptation to various systems, although challenges remain in handling highly nonlinear or multi-physics scenarios. Future work will focus on enhancing robustness, extending applicability, and integrating physical constraints, promising a new era of scalable, high-fidelity scientific modeling.
Deep Dive
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.