Mean-Flow Neural Operators for Long-Horizon Dynamical System Prediction
Keywords:
Neural Operators, Dynamical Systems, Long-Horizon Prediction, Scale SeparationAbstract
The prediction of complex dynamical systems over extended temporal horizons remains a fundamental challenge in computational science. Traditional numerical solvers often suffer from compounding discretization errors and prohibitive computational costs when simulating multiscale phenomena. Recently, neural operators have emerged as a promising paradigm for learning mesh-independent mappings between function spaces, yet they frequently struggle with error accumulation and spectral bias during autoregressive long-term rollouts. To address these limitations, this paper introduces the Mean-Flow Neural Operator, a novel architecture designed to explicitly decouple the macroscopic mean-flow dynamics from microscopic transient fluctuations. By embedding domain-agnostic scale separation directly into the operator learning framework, the proposed method preserves global conservation properties while dynamically modeling high-frequency residual interactions. Comprehensive empirical evaluations across diverse physical systems demonstrate that the Mean-Flow Neural Operator significantly attenuates long-horizon error propagation. Furthermore, the approach achieves substantial computational acceleration compared to high-fidelity numerical baselines. This work establishes a robust foundation for the deployment of data-driven surrogate models in critical applications requiring sustained predictive stability, ranging from climate modeling to advanced aerospace engineering.
References
1. Roh, S. B., Oh, S. K., Pedrycz, W., Wang, Z., Fu, Z., & Seo, K. (2022). Design of iterative fuzzy radial basis function neural networks based on iterative weighted fuzzy c-means clustering and weighted LSE estimation. IEEE Transactions on Fuzzy Systems, 30(10), 4273-4285.
2. Chen, Y., Lyu, N., Lang, S., Yan, H., Tao, Z., Ding, X., & Zhu, X. EconAI: Dynamic Persona Evolution and Memory-Aware Agents inEvolving Economic Environments. In Workshop on Multi-Agent Learning and Its Opportunities in the Era of Generative AI.
3. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B, 58(1), 267–288.
4. Wang, Jiacheng, et al. "A Dynamic Factor Gating Architecture with Market Regime Awareness for Stock Return Forecasting." (2026).
5. Dong, J., Koniusz, P., Chen, J., Xie, X., & Ong, Y. S. (2024). Adversarially robust few-shot learning via parameter co-distillation of similarity and class concept learners. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 28535-28544).
6. Qu, D., & Ma, Y. (2025, November). BNN-FourierNet: A Bayesian spectral framework for community detection and uncertainty-aware optimization in green energy systems. In 2025 6th International Conference on Artificial Intelligence and Computer Engineering (ICAICE) (pp. 932-936). IEEE.
7. Huang, J. (2025, September). GRAPHTRUST: GRAPH NEURAL NETWORKS FOR RELIABLE SUPPLY CHAIN RISK REASONING. In The 5th International scientific and practical conference “Trends in the development of science by young scientists and students”(September 30-October 03, 2025) Warsaw, Poland. International Science Group. 2025. 122 p. (p. 18).
8. Ma, Y., Qu, D., & Wang, Y. (2026). Dynamic community detection using class preserving time series generation with Fourier Markov diffusion. Scientific Reports.（https://doi.org/10.1038/s41598-026-37699-1）
9. Qu, D., & Ma, Y. (2026, March). SC-MGM: soft clustering and Markov-GNN modeling for spatiotemporal community detection in video scene graphs. In Ninth International Conference on Video and Image Processing (ICVIP 2025) (Vol. 14068, pp. 45-55). SPIE.
10. Kang, Hui, et al. "Unleashing the potential of regularization strategies in learning with noisy labels." arXiv preprint arXiv:2307.05025 (2023).
11. Wang, Z., Kim, E. H., Oh, S. K., Pedrycz, W., Fu, Z., & Yoon, J. H. (2024). Reinforced fuzzy-rule-based neural networks realized through streamlined feature selection strategy and fuzzy clustering with distance variation. IEEE Transactions on Fuzzy Systems, 32(10), 5674-5686.
12. Qu, D., & Ma, Y. (2025). Magnet-bn: markov-guided Bayesian neural networks for calibrated long-horizon sequence forecasting and community tracking. Mathematics, 13(17), 2740.
13. Yang, Tianyue, and Xiao Xue. "Meno: Meanflow-enhanced neural operators for dynamical systems." arXiv preprint arXiv:2604.06881 (2026).
14. Xue, Xiao, et al. "Fast-Forward Lattice Boltzmann: Learning Kinetic Behaviour with Physics-Informed Neural Operators." arXiv preprint arXiv:2509.22411 (2025).
15. Li, W. W., Ziser, Y., Xie, Y., Cohen, S. B., & Ma, T. (2025, July). Tsprank: Bridging pairwise and listwise methods with a bilinear travelling salesman model. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 1 (pp. 707-718).
16. Lu, Yao, Kaiyi Hu, and Luyan Zhang. "S 3 G: Stock State Space Graph for Enhanced Stock Trend Prediction." ICASSP 2026-2026 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2026.
17. Dong, J., Koniusz, P., Chen, J., & Ong, Y. S. (2024, September). Adversarially robust distillation by reducing the student-teacher variance gap. In European Conference on Computer Vision (pp. 92-111). Cham: Springer Nature Switzerland.
18. Dong, J., Moosavi-Dezfooli, S. M., Lai, J., & Xie, X. (2023). The enemy of my enemy is my friend: Exploring inverse adversaries for improving adversarial training. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 24678-24687).
19. Hao, Y., Xu, J., & Liao, S. (2026). RAPS-Net: A Risk-Aware CNN-LSTM Framework for Cross-Domain Risk Prediction and Dynamic Security Control in Cloud Payment Supply Chains. Economics and Management Innovation, 3(2), 18-27.
20. Dong, J., Moayedi, R. Z., Ong, Y. S., & Moosavi-Dezfooli, S. M. (2026). Allies teach better than enemies: Inverse adversaries for robust knowledge distillation. IEEE Transactions on Pattern Analysis and Machine Intelligence.
21. Xue, Xiao, et al. "Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model." arXiv preprint arXiv:2509.01293 (2025).
22. Li, W. W., Kim, H., Cucuringu, M., & Ma, T. (2026, August). Can LLM-based financial investing strategies outperform the market in long run?. In Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 1 (pp. 2711-2722).
23. Li, Z., Hao, Y., & Zeng, P. (2026). CLASNet: A Cognitive Load–Aware CNN-LSTM-Attention Framework for Supply Chain Demand Forecasting and Adaptive Human–Computer Interaction. Journal of Computer Science and Frontier Technologies, 3(2), 44-57.
24. Zhao, S., Qian, K., & Wen, C. (2022). Research on Used Car Transaction Cycle Based on Soft Voting. Information Systems and Economics, 3(1), 10-15.
25. Qu, D., & Ma, Y. (2026). F2-CommNet: Fourier–Fractional neural networks with Lyapunov stability guarantees for hallucination-resistant community detection. Frontiers in Computational Neuroscience, 19, 1731452. （https://doi.org/10.3389/fncom.2025.1731452）
26. Chen, L. (2026). Beyond external constraints: The missing dimension of ai governance. Available at SSRN 6449738.
27. Hu, Z., Hu, M., & Li, Z. (2026). DynSupplyNet: A Dynamic Graph Neural Network with Temporal Fusion for Supply Chain Risk Prediction and Propagation Analysis. Journal of Computer Science and Frontier Technologies, 3(2), 1-13.
28. Wu, Yuxuan, et al. "Unsupervised Disentanglement of Content and Style via Variance-Invariance Constraints." arXiv preprint arXiv:2407.03824 (2024).
29. Ma, Y., Qu, D., & Wang, Y. (2026). Tracking evolving communities in fake news cascades using temporal graphs. Scientific Reports.（https://www.nature.com/articles/s41598-026-35175-4）
30. Yao, Zhengtao, et al. "Task-agnostic federated learning." arXiv preprint arXiv:2406.17235 (2024).
31. Ma, Y., Qu, D., & Pyrozhenko, M. (2026). Bio-RegNet: A Meta-Homeostatic Bayesian Neural Network Framework Integrating Treg-Inspired Immunoregulation and Autophagic Optimization for Adaptive Community Detection and Stable Intelligence. Biomimetics, 11(1), 48.
32. Lo, S. C., Zingaro, A., McCullough, J. W., Xue, X., Gonzalez-Martin, P., Joo, B., ... & Coveney, P. V. (2025). A multi-component, multi-physics computational model for solving coupled cardiac electromechanics and vascular haemodynamics. Computer Methods in Applied Mechanics and Engineering, 446, 118185.
33. Xu, Y. (2020). A review of machine learning with echo state networks. Proj. Rep.
34. Lv, Qi, et al. "Spatial-temporal graph diffusion policy with kinematic modeling for bimanual robotic manipulation." Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.
35. Qu, D., Ma, Y., & Pyrozhenko, M. (2026). DISPEL-GNN: De-Illusion via Spectral Stability and Perturbation Bound-Enforced Learning for Community Detection with Risk-Aware Dynamic Attention in Graph Neural Networks. Mathematics, 14(4), 602.（https://doi.org/10.3390/math14040602）
36. Yang, C., Wang, Z., Oh, S. K., Pedrycz, W., & Yang, B. (2022). Ensemble fuzzy radial basis function neural networks architecture driven with the aid of multi-optimization through clustering techniques and polynomial-based learning. Fuzzy sets and systems, 438, 62-83.
37. Xing, J., Liu, Y., Zhang, Z., Chen, L., Hasan, M. N., & Zhang, Z. B. (2025). An Intrinsically Knowledge-Transferring Developmental Spiking Neural Network for Tactile Classification. IEEE Transactions on Neural Networks and Learning Systems.
Porter, M. E. (1985). Competitive advantage: Creating and sustaining superior performance. Free Press.
Wirtz, B. W., Weyerer, J. C., & Geyer, C. (2019). Artificial intelligence and the public sector—Applications and challenges. International Journal of Public Administration, 42(7), 596–615. https://doi.org/10.1080/01900692.2018.1498103
Doz, Y., & Kosonen, M. (2010). Embedding strategic agility: A leadership agenda for accelerating business model renewal. Long Range Planning, 43(2–3), 370–382. https://doi.org/10.1016/j.lrp.2009.07.006
Lee, I., & Shin, Y. J. (2018). Fintech: Ecosystem, business models, investment decisions, and challenges. Business Horizons, 61(1), 35–46. https://doi.org/10.1016/j.bushor.2017.09.003
Hart, S. L., & Dowell, G. (2011). A natural-resource-based view of the firm: Fifteen years after. Journal of Management, 37(5), 1464–1479. https://doi.org/10.1177/0149206310390219
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Seulgi Nam, Valentin Aubert
This work is licensed under a Creative Commons Attribution 4.0 International License.
