Research Article

Smoothing Screening and Reweighted Sparse Discovery of Noisy Governing Equations

Authors

Nadia Ghasemabadi * Department of Computer Engineering and Information Technology, La.C., Islamic Azad University, Lahijan, Iran.
Setayesh Zahiri Morvarid Intelligent Industrial Systems Research Group, Iran.
Fatemeh Rasoulpour Department of Computer Engineering and Information Technology, La.C., Islamic Azad University, Lahijan, Iran.

https://doi.org/10.22105/scfa.v3i2.106

Downloads

Abstract

Discovering governing equations from noisy trajectories depends on derivative estimation, library conditioning, and model selection. This manuscript proposes HERMES, a broader slow-fast and symbolic architecture, and evaluates a regression-side realization combining per-state Savitzky–Golay filtering, window selection, correlation screening, and reweighted thresholded least squares. The reported campaign uses Lotka–Volterra, unforced Duffing, and a three-species multiscale decay chain, with four noise levels and twenty realizations per system and level. At 5% noise the pooled reported exact-recovery rate is 0.667 versus 0.000 for the tested raw-finite-difference STLSQ and ensemble-lite baselines; coefficient relative error is 0.0551±0.0647 versus 0.4615±0.5710 for STLSQ. The chain has no reported exact recovery at any level. At 10% noise Duffing’s 0.900 rate corresponds to two failures among twenty realizations. The campaign does not test hidden-state recovery, neural slow-fast decomposition, evolutionary refinement, or a Pareto-front output. Aggregation units and statistical implementation require original records, and unequal derivative preprocessing limits attribution of the observed advantage to sparse-discovery methodology.

Keywords

Equation discovery Multiscale systems Symbolic regression Sparse identification Slow-fast dynamics Noise robustness

References (37)

[1] Michaelis, L., & Menten, M. L. (1913). Die kinetik der invertinwirkung. Biochem. z, 49(333–369), 352. https://www.google.com/goto?url=CAESkwEBxCJF1wK-GOVfJ3ezRkPV2vcfbNziyq4F-1yzdwHUvDMVf9YvEA1zynTMVR4yzQnT4A71alHCWQyaEdC7xsKynoP9_c4h-ufe4iDkDNdDXVlV-BpbTB06qFdC_CO_igeVLB-ytytxObddbn_jrzXxdNab27g-dFkL3a5DOrDuOQNOXDu4VjyVcqATnZjgUBVkIdA

[2] Bongard, J., & Lipson, H. (2007). Automated reverse engineering of nonlinear dynamical systems. Proceedings of the national academy of sciences, 104(24), 9943–9948. https://doi.org/10.1073/pnas.0609476104

[3] Schmidt, M., & Lipson, H. (2009). Distilling free-form natural laws from experimental data. Science, 324(5923), 81–85. https://doi.org/10.1126/science.1165893

[4] Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15), 3932–3937. https://doi.org/10.1073/pnas.1517384113

[5] Chen, R. T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. K. (2018). Neural ordinary differential equations. Advances in neural information processing systems, 31. https://proceedings.neurips.cc/paper_files/paper/2018/hash/69386f6bb1dfed68692a24c8686939b9-Abstract.html

[6] Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. Journal of differential equations, 31(1), 53–98. https://doi.org/10.1016/0022-0396(79)90152-9

[7] Verhulst, F. (2005). Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics. Springer. https://www.amazon.com/dp/1441919929?lv=shuf&channelId=500&plpRedirect=mhFallback

[8] Takens, F. (2006). Detecting strange attractors in turbulence. Dynamical systems and turbulence, warwick 1980: proceedings of a symposium held at the university of warwick 1979/80 (pp. 366–381). Springer. https://doi.org/10.1007/BFb0091924

[9] Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science advances, 3(4), e1602614. https://doi.org/10.1126/sciadv.1602614

[10] Udrescu, S.-M., & Tegmark, M. (2020). AI Feynman: A physics-inspired method for symbolic regression. Science advances, 6(16), eaay2631. https://doi.org/10.1126/sciadv.aay2631

[11] Cranmer, M. (2023). Interpretable machine learning for science with PySR and symbolicregression. https://doi.org/10.48550/arXiv.2305.01582

[12] Zheng, P., Askham, T., Brunton, S. L., Kutz, J. N., & Aravkin, A. Y. (2018). A unified framework for sparse relaxed regularized regression: SR3. IEEe access, 7, 1404–1423. https://doi.org/10.1109/ACCESS.2018.2886528

[13] Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019). Data-driven discovery of coordinates and governing equations. Proceedings of the national academy of sciences, 116(45), 22445–22451. https://doi.org/10.1073/pnas.1906995116

[14] De Silva, B. M., Champion, K., Quade, M., Loiseau, J. C., Kutz, J. N., & Brunton, S. L. (2020). Pysindy: A python package for the sparse identification of nonlinear dynamics from data. https://doi.org/10.48550/arXiv.2004.08424

[15] Kaiser, E., Kutz, J. N., & Brunton, S. L. (2018). Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proceedings mathematical, physical, and engineering sciences, 474(2219), 20180335. https://doi.org/10.1098/rspa.2018.0335

[16] Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., … & Edelman, A. (2020). Universal differential equations for scientific machine learning. https://doi.org/10.48550/arXiv.2001.04385

[17] Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of computational physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045

[18] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature reviews physics, 3(6), 422–440. https://doi.org/10.1038/s42254-021-00314-5%0A%0A

[19] Greydanus, S., Dzamba, M., & Yosinski, J. (2019). Hamiltonian neural networks. Advances in neural information processing systems (Vol. 32). NeurIPS Proceedings. https://proceedings.neurips.cc/paper_files/paper/2019/hash/26cd8ecadce0d4efd6cc8a8725cbd1f8-Abstract.html

[20] Cranmer, M., Greydanus, S., Hoyer, S., Battaglia, P., Spergel, D., & Ho, S. (2020). Lagrangian neural networks. https://doi.org/10.48550/arXiv.2003.04630

[21] Schmid, P. J. (2010). Dynamic mode decomposition of numerical and experimental data. Journal of fluid mechanics, 656, 5–28. https://doi.org/10.1017/S0022112010001217

[22] Chartrand, R. (2011). Numerical differentiation of noisy, nonsmooth data. International scholarly research notices, 2011(1), 164564. https://doi.org/10.5402/2011/164564

[23] Huber, P. J. (1992). Robust estimation of a location parameter. In Breakthroughs in statistics: methodology and distribution (pp. 492–518). Springer. https://doi.org/10.1007/978-1-4612-4380-9_35%0A%0A

[24] Meinshausen, N., & Bühlmann, P. (2010). Stability selection. Journal of the royal statistical society: Series b (statistical methodology), 72(4), 417–473. https://doi.org/10.1111/j.1467-9868.2010.00740.x

[25] Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716–723. https://doi.org/10.1109/TAC.1974.1100705

[26] Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics, 461–464. https://www.jstor.org/stable/2958889

[27] Lotka, A. J. (1998). Analytical theory of biological populations. Springer Science & Business Media. https://doi.org/10.1007/978-1-4757-9176-1

[28] Duffing, G. (1918). Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg. https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp106773

[29] Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634

[30] Fasel, U., Kutz, J. N., Brunton, B. W., & Brunton, S. L. (2022). Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proceedings mathematical, physical, and engineering sciences, 478(2260), 20210904. https://doi.org/10.1098/rspa.2021.0904

[31] Savitzky, A., & Golay, M. J. E. (1964). Smoothing and differentiation of data by simplified least squares procedures. Analytical chemistry, 36(8), 1627–1639. https://doi.org/10.1021/ac60214a047

[32] Golub, G. H., Heath, M., & Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21(2), 215–223. https://doi.org/10.1080/00401706.1979.10489751

[33] Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., & Cournapeau, D. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature methods, 17(3), 261–272. https://doi.org/10.1038/s41592-019-0686-2

[34] Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in science & engineering, 9(3), 90–95. https://doi.org/10.1109/MCSE.2007.55

[35] Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics bulletin, 1, 80. https://doi.org/10.2307/3001968

[36] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, 6(2), 65–70. https://www.jstor.org/stable/4615733

[37] Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the american statistical association, 32, 675–701. https://doi.org/10.1080/01621459.1937.10503522

Article Metrics

Published 2026-06-17
Issue Vol. 3 No. 2 (2026): Soft Computing Fusion with Applications (SCFA)
Section Articles
Document Type Research Article
Views 0
Downloads 0

How to Cite

Ghasemabadi, N. ., Zahiri, S. ., & Rasoulpour, F. . (2026). Smoothing Screening and Reweighted Sparse Discovery of Noisy Governing Equations. Soft Computing Fusion With Applications , 3(2), 139-152. https://doi.org/10.22105/scfa.v3i2.106

Download Citation

Similar Articles

Archit Awasthi, Pradeep Kumar Singh, Akhand Pratap Shukla, Bridging AI and Cryptography for Robust Security , Soft Computing Fusion with Applications : Vol. 1 No. 4 (2024): Soft Computing Fusion with Applications
Daniel Dasig Jr, A Fuzzy Set-Based Context-Aware Decision Framework for Histopathological Image Classification in Tumor Microarrays , Soft Computing Fusion with Applications : Vol. 2 No. 2 (2025): Soft Computing Fusion with Applications
Mohammad Sheihan Javaid, Deep Learning-Based Segmentation for Land Management Using Satellite Imagery , Soft Computing Fusion with Applications : Vol. 1 No. 4 (2024): Soft Computing Fusion with Applications
Jyotsana Grover, Shakila Devi G T, Communication-Efficient Federated Rare-Pattern Mining under Non-IID Data and Privacy Constraints , Soft Computing Fusion with Applications : Vol. 3 No. 2 (2026): Soft Computing Fusion with Applications (SCFA)
Ismail A Mageed, Hong Li, Reaching The Pinnacle of the Digital WorldOpen Mathematical Problems in AI-Driven Robot Control , Soft Computing Fusion with Applications : Vol. 2 No. 2 (2025): Soft Computing Fusion with Applications
Kaosar Hossain, Harnessing Quantum Computing and Artificial Intelligence for Next-Generation Cybersecurity: A Comprehensive Review , Soft Computing Fusion with Applications : Vol. 2 No. 4 (2025): Soft Computing Fusion with Applications
Mojtaba Nasehi, An Explainable AI–Driven Type-2 Fuzzy Decision Support Framework for Sustainable Energy Management in Smart Grids , Soft Computing Fusion with Applications : Vol. 3 No. 1 (2026): Soft Computing Fusion with Applications
Chukwuemeka Joshua Okafor, Adedayo Ayomide Adeniran, Adetayo Olaniyi Adeniran, Machine Learning and AI for Predictive Maintenance and Grid Integration of Wind Farms , Soft Computing Fusion with Applications : Vol. 2 No. 2 (2025): Soft Computing Fusion with Applications
Omid Eslamifar, Mohammadreza Soltani, Seyed Mohammad Jalal Rastegar Fatemi, Multiclass Blood Cell Classification Using Contourlet Transform and Metaheuristic-Optimized Deep Features with Clustering-Based Decision Making , Soft Computing Fusion with Applications : Vol. 1 No. 4 (2024): Soft Computing Fusion with Applications
Shamina Israt Tithi, Kuan Mo, Md Amran Hossen Pabel, Tahia Tasnuva, Md Omar Farukh, AI-Driven Sustainability or Carbon Trap? Rethinking Energy Use and FDI in U.S. Environmental Performance , Soft Computing Fusion with Applications : Vol. 3 No. 1 (2026): Soft Computing Fusion with Applications

1-10 of 58 Next

You may also start an advanced similarity search for this article.