Rotasyonlu Termal Sığ Su Denklemi için Çekirdek Temel Bileşen Analizi ile Doğrusal Olmayan Mertebe İndirgenmiş Modelleme

Öz

Bu çalışmada, rotasyonlu termal sığ su denklemi için çekirdek temel bileşen analizi (ÇTBA) tabanlı veriye dayalı doğrusal olmayan bir mertebe indirgenmiş modelleme yaklaşımı geliştirilmiştir. Yüksek boyutlu tam model, denklemlerin uzayda sonlu farklar yöntemi ile ayrıklaştırılması ve zamanda enerji korumalı ortalama vektör alanı yöntemi ile çözülmesi sonucunda elde edilmiştir. Önerilen yöntem, çözüm manifoldunun doğrusal olmayan düşük boyutlu temsilini elde etmek için ÇTBA kullanmaktadır. ÇTBA yönteminde, çekirdek fonksiyonlar aracılığıyla anlık görüntü verileri örtük olarak yüksek boyutlu bir özellik uzayına taşınmakta ve doğrusal altuzay yöntemlerine kıyasla doğrusal olmayan tutarlı yapılar daha etkin biçimde yakalanabilmektedir. Önerilen yöntem doğrusal olmayan jeofizik akış modellerinin etkin benzetimi için umut verici bir yaklaşım oluşturmaktadır.

Anahtar Kelimeler

• Rotasyonlu termal sığ su denklemi
• çekirdek temel bileşenler analizi
• mertebe indirgenmiş modelleme

Nonlinear Reduced-Order Modeling of the Rotating Thermal Shallow Water Equation via Kernel Principal Component Analysis

Öz

In this study, a data-driven nonlinear reduced-order modeling framework based on Kernel Principal Component Analysis (KPCA) is developed for the rotating thermal shallow water equation. The high-dimensional full-order model is obtained by discretizing the governing equations in space using finite differences and integrating in time with the energy-preserving average vector field method. The proposed approach employs KPCA to construct a nonlinear low-dimensional representation of the solution manifold, by implicitly mapping the snapshot data into a high-dimensional feature space through kernel functions, and captures nonlinear coherent structures more effectively than the linear subspace methods. The proposed framework is a promising approach for efficient simulation of nonlinear geophysical flow models.

Anahtar Kelimeler

• Rotating thermal shallow water equation
• kernel principal component analysis
• reduced-order modeling

Destekleyen Kurum

The authors declare that no financial support was received for the research, authorship, or publication of this study.

Etik Beyan

The authors declare that this study does not require any ethics committee approval or special permission.

Kaynakça

1. Eldred, C., Dubos, T., & Kritsikis, E. (2019). A quasi-Hamiltonian discretization of the thermal shallow water equations. Journal of Computational Physics, 379, 1–31. https://doi.org/10.1016/j.jcp.2018.10.038
2. Ripa, P. (1995). On improving a one-layer ocean model with thermodynamics. Journal of Fluid Mechanics, 303, 169–201. https://doi.org/10.1017/S0022112095004228
3. Warneford, E. S., & Dellar, P. J. (2013). The quasi-geostrophic theory of the thermal shallow water equations. Journal of Fluid Mechanics, 723, 374–403. https://doi.org/10.1017/jfm.2013.101
4. Afkham, B. M., & Hesthaven, J. (2017). Structure preserving model reduction of parametric Hamiltonian systems. SIAM Journal on Scientific Computing, 39(6), A2616–A2644. https://doi.org/10.1137/17M1111991
5. Peng, L., & Mohseni, K. (2016). Symplectic model reduction of Hamiltonian systems. SIAM Journal on Scientific Computing, 38(1), A1–A27. https://doi.org/10.1137/140978922
6. Karasözen, B., Yıldız, S., & Uzunca, M. (2021). Structure preserving model order reduction of shallow water equations. Mathematical Methods in the Applied Sciences, 44(1), 476–492. https://doi.org/10.1002/mma.6751
7. Barrault, M., Maday, Y., Nguyen, N. C., & Patera, A. T. (2004). An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, 339(9), 667–672. https://doi.org/10.1016/j.crma.2004.08.006
8. Chaturantabut, S., & Sorensen, D. C. (2010). Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5), 2737–2764. https://doi.org/10.1137/090766498

9. Antil, H., Heinkenschloss, M., & Sorensen, C., Danny (2014). Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. In A. Quarteroni & G. Rozza, editors, Reduced Order Methods for Modeling and Computational Reduction, volume 9 of MS & A - Modeling, Simulation and Applications, Springer International Publishing, 101–136, https://doi.org/10.1007/978-3-319-02090-7_4
10. Cohen, D., & Hairer, E. (2011). Linear energy-preserving integrators for Poisson systems. BIT Numerical Mathematics, 51(1), 91–101. https://doi.org/10.1007/s10543-011-0310-z
11. Berkooz, G., Holmes, P., & Lumley, J. L. (1993). The proper orthogonal decomposition in the analysis of turbulent flows. Annual review of fluid mechanics, 25(1), 539–575. https://doi.org/10.1146/annurev.fl.25.010193.002543
12. Alla, A., & Kutz, J. N. (2017). Nonlinear model order reduction via dynamic mode decomposition. SIAM Journal on Scientific Computing, 39(5), B778–B796. https://doi.org/10.1137/16M1059308
13. Schölkopf, B., Smola, A., & Müller, K. R. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10, 1299–1319. https://doi.org/10.1162/089976698300017467
14. Mika, S., Schölkopf, B., Smola, A., Müller, K. R., Scholz, M., & Rätsch, G. (1998). Kernel pca and de-noising in feature spaces. In M. Kearns, S. Solla, & D. Cohn, editors, Advances in Neural Information Processing Systems, volume 11, MIT Press, 536–542.
15. Schölkopf, B., & Smola, A. (2002). Learning with kernels: support vector machines, regularization, optimization, and beyond. Adaptive computation and machine learning, MIT Press, Cambridge, Mass.
16. Shawe-Taylor, J., & Cristianini, N. (2004). Kernel Methods for Pattern Analysis. Cambridge University Press, https://doi.org/10.1017/CBO9780511809682
17. Muller, K. R., Mika, S., Ratsch, G., Tsuda, K., & Scholkopf, B. (2001). An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2), 181–201. https://doi.org/10.1109/72.914517
18. Wang, Q. (2012). Kernel principal component analysis and its applications in face recognition and active shape models. Computer Vision and Pattern Recognition, 1–8. https://doi.org/10.48550/arXiv.1207.3538
19. Cakır, Y., & Uzunca, M. (2024). Kernel principal component analysis for Allen-Cahn equations. Mathematics, 12, 3434. https://doi.org/10.3390/math12213434
20. Vapnik, V. N. (1998). Statistical Learning Theory. Michael Jordan Department of Computer Science University of Califomia, Berkeley, Springer.
21. Rathi, Y., Dambreville, S., & Tannenbaum, A. (2006). Statistical shape analysis using kernel PCA. School of Electrical and Computer Engineering Georgia Institute of Technology, 6064, 1–8. https://doi.org/10.1117/12.64141715
22. Qian, E., Kramer, B., Marques, A. N., & Willcox, K. E. (2019). Transform & Learn: A data-driven approach to nonlinear model reduction. Verification, Validation, and Uncertainty Quantification. AIAA, 1–11, https://doi.org/10.2514/6.2019-3707
23. 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
24. Kaiser, E., Kutz, J. N., & Brunton, S. L. (2020). Data-driven approximations of dynamical systems operators for control. In The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications, Springer International Publishing, Cham, 197–234, https://doi.org/10.1007/978-3-030-35713-9_8
25. Swischuk, R., Mainini, L., Peherstorfer, B., & Willcox, K. (2019). Projection-based model reduction: Formulations for physics-based machine learning. Computers & Fluids, 179, 704–717. https://doi.org/10.1016/j.compfluid.2018.07.021
26. Freno, B. A., & Carlberg, K. T. (2019). Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations. Computer Methods in Applied Mechanics and Engineering, 348, 250–296. https://doi.org/10.1016/j.cma.2019.01.024
27. Salmon, R. (2004). Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations. Journal of the Atmospheric Sciences, 61(16), 2016–2036. https://doi.org/10.1175/1520-0469(2004)061<2016:PATTCO>2.0.CO;2
28. Çakır, Y., & Uzunca, M. (2024). Nonlinear reduced order modelling for Korteweg-de Vries equation. International Journal of Informatics and Applied Mathematics, 7, 57–72. https://doi.org/10.53508/ijiam.1455321
29. Karasözen, B., Uzunca, M., & Küçükseyhan, T. (2015). Structure preserving integration and model order reduction of skew-gradient reaction-diffusion systems. Annals of Operations Research, 258, 79–106. https://doi.org/10.1007/s10479-015-2063-6
30. Buchfink, P., Bhatt, A., & Haasdonk, B. (2019). Symplectic model order reduction with non-orthonormal bases. Mathematical & Computational Applications, 24(2), Paper No. 43, 26. https://doi.org/10.3390/mca24020043
31. Micchelli, C. A. (1986). Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constructive Approximation, 2, 11–22. https://doi.org/10.1007/BF01893414
32. Buhmann, M. D. (2003). Radial Basis Functions: Theory and Implementations. Justus-Liebig-Universität Giessen, Germany, Cambridge University Press, https://doi.org/10.1017/CBO9780511543241
33. Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. The MIT Press Cambridge Massachusetts.
34. [34] Aydin, D. (2018). Alternative robust estimation methods for parameters of Gumbel distribution: An application to wind speed data with outliers. Wind Struct, 26, 383–395. https://doi.org/10.12989/WAS.2018.26.6.383
35. [35] Aydin, D., & Senoglu, B. (2015). Monte Carlo comparison of the parameter estimation methods for the two-parameter Gumbel distribution. J Mod Appl Stat Method, 14, 123–140. https://doi.org/10.22237/jmasm/1446351060
36. [36] Tiku, M. L., & Akkaya, A. D. (2004). Robust Estimation and Hypothesis Testing. Springer Series in Computational Mathematics, New Age International: New Delhi, India.
37. [37] Cox, T. F., & Cox, M. A. A. (2001). Multidimensional Scaling. Monographs on Statistics and Applied Probability, Chapman & Hall, London, U.K.