Spectral-Residual Koopman Representation Learning and Semigroup Estimation for Stochastic Dynamical Systems under Noisy Observations and Distributional Variability
Keywords:
Koopman Operator, Stochastic Dynamical Systems, Spectral-Residual Learning, Semigroup EstimationAbstract
The study of complex stochastic dynamical systems is of paramount importance across a multitude of scientific disciplines ranging from fluid mechanics and climate modeling to power grid stabilization and financial forecasting. Traditional approaches rely heavily on nonlinear differential equations which often become intractable when subjected to high-dimensional state spaces and intricate stochastic forcing. The Koopman operator theory provides an elegant alternative by lifting these finite-dimensional nonlinear dynamics into an infinite-dimensional space where their evolution becomes strictly linear. However, practical implementations of Koopman theory via data-driven representation learning face severe limitations when observational data is corrupted by noise or when the underlying data-generating distributions exhibit non-stationarity and variability. This paper proposes a novel framework termed Spectral-Residual Koopman Representation Learning. By integrating spectral residual regularization into a deep learning-based representation framework, the proposed methodology explicitly enforces robust semigroup estimation even under substantial observational noise. Furthermore, invariant risk minimization techniques are adapted to ensure that the learned observables remain consistent across varying environmental distributions. Through rigorous theoretical analysis and extensive empirical evaluations on both synthetic benchmark systems and simulated real-world fluid dynamics, the proposed approach demonstrates significant improvements in long-term predictive accuracy and spectral consistency over existing baseline methods.
References
1. Xiaoteng Ma, Yiqin Yang, Hao Hu, Qihan Liu, Jun Yang, Chongjie Zhang, Qianchuan Zhao, and Bin Liang. Offline reinforcement learning with value-based episodic memory. arXiv preprint arXiv:2110.09796, 2021b.
2. Qu, D., Ma, Y., & Wang, Y. (2026). STR-GNN: stability-regularized graph neural networks for suppressing spurious temporal variations in dynamic community detection. Scientific Reports.
3. Ma, Y., & Qu, D. (2026, February). IMFM-GNN: Interpretable Multimodel GNN Via Fourier-Markov Dynamics for Community Detection in Sustainable Energy Networks. In 2026 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA) (pp. 1-6). IEEE.
4. Luo, T., Li, Q., Jiang, M., & Hou, C. (2025). Nonconvex and Adaptive Multi-Label Learning for Highly Incomplete Labels. Knowledge-Based Systems, 114784.
5. Xu, Y. (2020). A review of machine learning with echo state networks. Proj. Rep.
6. Wu, Beiliang, et al. "IndexNet: Timestamp and Variable-Aware Modeling for Time Series Forecasting." arXiv preprint arXiv:2509.23813 (2025).
7. Fang, R., Zhao, J., Wang, S., Pu, R., Li, B., Cai, J., ... & Wang, B. (2025). Saga: Structural aggregation guided alignment with dynamic view and neighborhood order selection for multiview graph domain adaptation. In The Fourteenth International Conference on Learning Representations.
8. Han, Z., Chen, W., & Han, Y. (2024, August). Data Quality-Driven Top-k Rule Discovery. In International Conference on Machine Learning, Cloud Computing and Intelligent Mining (pp. 24-38). Singapore: Springer Nature Singapore.
9. Kingma, D. P., & Ba, J. (2015). Adam: A method for stochastic optimization. In Proceedings of ICLR.
10. Zhu, R., Jiang, Z., Wu, J., Ma, Z., Song, J., Bai, F., ... & He, C. (2025, April). Grait: Gradient-driven refusal-aware instruction tuning for effective hallucination mitigation. In Findings of the Association for Computational Linguistics: NAACL 2025 (pp. 4006-4021).
11. Shor, P. W. (1997). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5), 1484–1509.
12. Lv, Q., Deng, X., Chen, G., Wang, M. Y., & Nie, L. (2024). Decision mamba: A multi-grained state space model with self-evolution regularization for offline rl. Advances in neural information processing systems, 37, 22827-22849.
13. Wei, W. D., Fang, H. B., Liu, Y. D., Shi, J. X., Kwok, J., & Li, Y. F. (2026). Activation Compression in LLMs: Theoretical Analysis and Efficient Algorithm. arXiv preprint arXiv:2605.01255.
14. Wang, J., & Al Faruque, M. A. (2025, April). Transformer-based contrastive meta-learning for low-resource generalizable activity recognition. In ICASSP 2025-2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)(pp. 1-5). IEEE.
15. Gu, C., Zhang, W., Huang, Z., Kou, J., Liu, Z., Zhao, C., Liu, C., Zhang, L., Lin, W., Wang, Z., Deng, J., Xie, Y., Huang, G., Zhang, C., Lu, X., Wang, C., Zhang, Z., Yuan, H., Duan, X., & Fang, Y. (2024). LENS: Layers of evaluation of hallucination in GenAI systems. In 2024 7th International Conference on Universal Village (UV) (pp. 1–85). IEEE. https://doi.org/10.1109/UV63228.2024.11189150
16. Hu, H., Yang, Y., Zhao, Q., & Zhang, C. (2022, June). On the role of discount factor in offline reinforcement learning. In International conference on machine learning (pp. 9072-9098). PMLR.
17. Zhang, Y., Zhang, Y., Zhao, M., Feng, S., & Cheung, Y. M. (2026). How to Ask the AI: A User Perspective Survey for Large Language Model Prompting.IEEE Transactions on Artificial Intelligence.
18. Xue, X., Feng, L., Feng, Y., Liu, R., Zhang, K., & Tan, K. C. (2025, June). A theoretical analysis of analogy-based evolutionary transfer optimization. In2025 IEEE Congress on Evolutionary Computation (CEC)(pp. 1-8). IEEE.
19. Ma, Y., Qu, D., & Wang, Y. (2026). HALO-GNN: hallucination-resistant temporal graph neural networks for dynamic community detection.Scientific Reports.
20. Pei, Zhengqi, Qingming Huang, and Shuhui Wang. "When LLMs develop languages: Symbolic communication for efficient multi-agent reasoning." arXiv preprint arXiv:2606.29354 (2026).
21. Cao, C., Zhang, K., Xue, X., Tan, K. C., Wang, J., Zhang, L., ... & Yan, X. (2024). Global and local search experience-based evolutionary sequential transfer optimization.IEEE Transactions on Evolutionary Computation,29(4), 1269-1283.
22. Jin, X., Diao, L., Xiao, Q., Hu, Y., Zhang, Z., Liu, Y., & Gu, H. (2025). CCAD: Compressed Global Feature Conditioned Anomaly Detection. arXiv preprint arXiv:2512.21459.
23. Xue, X., Zhang, K., Tan, K. C., Feng, L., Wang, J., Chen, G., ... & Yao, J. (2020). Affine transformation-enhanced multifactorial optimization for heterogeneous problems.IEEE Transactions on Cybernetics,52(7), 6217-6231.
24. Zhu, R., Jiang, B., Mei, L., Yang, F., Wang, L., Gao, H., ... & Zhang, D. (2025). AdaptFlow: Adaptive Workflow Optimization via Meta-Learning. arXiv preprint arXiv:2508.08053.
25. Han, Z., Chen, W., Han, Y., Mao, R., & Qin, J. (2026). Fast Diversified Top-k Rule Discovery via User-Guided Embeddings. IEEE Transactions on Knowledge and Data Engineering.
26. Yang, Y., Ma, X., Li, C., Zheng, Z., Zhang, Q., Huang, G., ... & Zhao, Q. (2021). Believe what you see: Implicit constraint approach for offline multi-agent reinforcement learning. Advances in Neural Information Processing Systems, 34, 10299-10312.
27. Wang, J., & Faruque, M. A. A. (2023). Robust and scalable hyperdimensional computing with brain-like neural adaptations.arXiv preprint arXiv:2311.07705.
28. Liu, Chenchen, Fuqiang Liu, and Hai Li. "Beyond CMOS: memristor and its application for next generation storage and computing." Electrochemical Society Transactions 233 85, no. 6 (2018): 115-125.
29. Qu, D., Ma, Y., Yan, J., & Pyrozhenko, M. (2026). TriMeta-BFNet: A Tri-Meta Stacked Atypical-Frequency Bayesian Fourier Neural Network for Hallucination-Resistant Community Detection. Mathematics, 14 (13), 2283.
30. Zhang, J., Shi, Y., Ma, Y., Xu, L., Yu, J., & Wang, J. (2023, June). Ikol: Inverse kinematics optimization layer for 3d human pose and shape estimation via gauss-newton differentiation. In Proceedings of the AAAI conference on artificial intelligence (Vol. 37, No. 3, pp. 3454-3462).
31. Wang, J., Huang, S., & Imani, M. (2023, July). Disthd: A learner-aware dynamic encoding method for hyperdimensional classification. In2023 60th ACM/IEEE Design Automation Conference (DAC)(pp. 1-6). IEEE.
32. Yang Y, Hu H, Mao Y, et al. OPRIDE: Offline Preference-based Reinforcement Learning via In-Dataset Exploration[J]. arXiv preprint arXiv:2604.02349, 2026.
33. Liang, R., Xu, S., Xie, C., Chen, J., Ren, F., Yang, S., & Yabe, T. (2026). Abstain Mask Retain Core: Time Series Prediction by Adaptive Masking Loss with Representation Consistency. Advances in Neural Information Processing Systems, 38, 99445-99471.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Paul Mok, Hiu-Tung Yau
This work is licensed under a Creative Commons Attribution 4.0 International License.
