Kernel methods for center manifold approximation and a weak data-based version of the center manifold theorem. (English) Zbl 07477802

Summary: For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold.
Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system.
In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics is quantified.
We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.


34C45 Invariant manifolds for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations


Full Text: DOI arXiv


[1] Carr, J., Applications of Centre Manifold Theory, Volume 35 of Applied Mathematical Sciences (1981), Springer-Verlag: Springer-Verlag New York-Berlin · Zbl 0464.58001
[2] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Volume 840 of Lecture Notes in Mathematics (1981), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0456.35001
[3] Kelley, A., Stability of the center-stable manifold, J. Math. Anal. Appl., 18, 2, 336-344 (1967) · Zbl 0166.08304
[4] Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3, 546-570 (1967) · Zbl 0173.11001
[5] Pliss, V. A., A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28, 1297-1324 (1964) · Zbl 0131.31505
[6] Shoshitaishvili, A. N., Bifurcations of topological type of singular points of vector fields that depend on parameters, Funkcional. Anal. I Priložen., 6, 2, 97-98 (1972) · Zbl 0274.34028
[7] Haasdonk, B.; Hamzi, B.; Santin, G.; Wittwar, D., Greedy kernel methods for center manifold approximation, (Sherwin, S. J.; Moxey, D.; Peiró, J.; Vincent, P. E.; Schwab, C., Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 (2020), Springer International Publishing: Springer International Publishing Cham), 95-106 · Zbl 1485.37072
[8] Aronszajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 68, 337-404 (1950) · Zbl 0037.20701
[9] Alexander, R.; Giannakis, D., Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques, Physica D, 409, Article 132520 pp. (2020) · Zbl 1496.37085
[10] Bittracher, A.; Klus, S.; Hamzi, B.; Koltai, P.; Schütte, C., Dimensionality reduction of complex metastable systems via kernel embeddings of transition manifolds (2019), https://arxiv.org/abs/1904.08622
[11] J. Bouvrie, B. Hamzi, Balanced reduction of nonlinear control systems in Reproducing Kernel Hilbert Space, in: Proc. 48th Annual Allerton Conference on Communication, Control, and Computing, 2010, pp. 294-301.
[12] Bouvrie, J.; Hamzi, B., Kernel methods for the approximation of nonlinear systems, SIAM J. Control Optim., 55, 4, 2460-2492 (2017) · Zbl 1368.93248
[13] Bouvrie, J.; Hamzi, B., Kernel methods for the approximation of some key quantities of nonlinear systems, J. Comput. Dyn, 1 (2017) · Zbl 1394.37115
[14] Brünnette, T.; Santin, G.; Haasdonk, B., Greedy kernel methods for accelerating implicit integrators for parametric ODEs, (Radu, F. A.; Kumar, K.; Berre, I.; Nordbotten, J. M.; Pop, I. S., Numerical Mathematics and Advanced Applications - ENUMATH 2017 (2019), Springer International Publishing: Springer International Publishing Cham), 889-896 · Zbl 07136774
[15] R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, G. Santin, Approximating basins of attraction for dynamical systems via stable radial bases, in: AIP Conf. Proc., 2016.
[16] Giesl, P.; Hamzi, B.; Rasmussen, M.; Webster, K., Approximation of Lyapunov functions from noisy data, J. Comput. Dyn (2019)
[17] Hamzi, B.; Owhadi, H., Learning dynamical systems from data: a simple cross-validation perspective (2020), https://arxiv.org/abs/2007.05074
[18] Klus, S.; Nuske, F.; Hamzi, B., Kernel-based approximation of the Koopman generator and Schrödinger operator, Entropy, 22 (2020)
[19] Klus, S.; Nüske, F.; Peitz, S.; Niemann, J.-H.; Clementi, C.; Schütte, C., Data-driven approximation of the koopman generator: Model reduction, system identification, and control, Physica D, 406, Article 132416 pp. (2020) · Zbl 1485.93097
[20] Khalil, H. K., Nonlinear Systems, Volume 3 (1996), Prentice-Hall
[21] Lin, Y.; Sontag, E. D.; Wang, Y., A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34, 1, 124-160 (1996) · Zbl 0856.93070
[22] Wendland, H., Scattered Data Approximation, Volume 17 of Cambridge Monographs on Applied and Computational Mathematics (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1075.65021
[23] Micchelli, C. A.; Pontil, M., On learning vector-valued functions, Neural Comput., 17, 1, 177-204 (2005) · Zbl 1092.93045
[24] Wittwar, D.; Santin, G.; Haasdonk, B., Interpolation with uncoupled separable matrix-valued kernels, Dolomites Res. Notes Approx., 11, 23-29 (2018)
[25] Wittwar, D., Approximation with Matrix-Valued Kernels and Highly Effective Error Estimators for Reduced Basis Approximations (2020), Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, (Ph.D. thesis)
[26] De Marchi, S.; Schaback, R.; Wendland, H., Near-optimal data-independent point locations for radial basis function interpolation, Adv. Comput. Math., 23, 3, 317-330 (2005) · Zbl 1070.65008
[27] Wittwar, D.; Haasdonk, B., Greedy algorithms for matrix-valued kernels, (Radu, F. A.; Kumar, K.; Berre, I.; Nordbotten, J. M.; Pop, I. S., Numerical Mathematics and Advanced Applications ENUMATH 2017 (2019), Springer International Publishing: Springer International Publishing Cham), 113-121 · Zbl 07136698
[28] Santin, G.; Haasdonk, B., Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation, Dolomites Res. Notes Approx., 10, 68-78 (2017) · Zbl 1370.94401
[29] Wenzel, T.; Santin, G.; Haasdonk, B., A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability and uniform point distribution, J. Approx. Theory, 262, Article 105508 pp. (2021) · Zbl 1458.65018
[30] Santin, G.; Haasdonk, B., Kernel methods for surrogate modeling, (Benner, P.; Grivet-Talocia, S.; Quarteroni, A.; Rozza, G.; Schilders, W.; Silveira, L. M., Model Order Reduction, Volume 2 (2021), De Gruyter)
[31] Fasshauer, G. E.; McCourt, M., Kernel-Based Approximation Methods using MATLAB, Volume 19 of Interdisciplinary Mathematical Sciences (2015), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ
[32] Wendland, H.; Rieger, C., Approximate interpolation with applications to selecting smoothing parameters, Numer. Math., 101, 4, 729-748 (2005) · Zbl 1081.65016
[33] Roberts, A. J., Construct centre manifolds of ordinary or delay differential equations (autonomous) (2020), http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php
[34] Driscoll, T.; Fornberg, B., Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl., 43, 3, 413-422 (2002) · Zbl 1006.65013
[35] Larsson, E.; Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49, 1, 103-130 (2005) · Zbl 1074.41012
[36] Roberts, A. J., Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations, The J. Australian Math. Soc. Ser. B. Appl. Math., 27, 1, 48-65 (1985) · Zbl 0576.76043
[37] Roberts, A. J., Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems (2019), https://arxiv.org/abs/1804.06998
[38] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4, 1, 389-396 (1995) · Zbl 0838.41014
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