×

Data-driven model reduction and transfer operator approximation. (English) Zbl 1396.37083

The authors establish connections between different data-driven model reduction and transfer operator approximation methods developed independently by the dynamical systems, fluid dynamics, machine learning and molecular dynamics communities. Although the derivations of these methods differ, it is shown that the resulting algorithms share many similarities.

MSC:

37M10 Time series analysis of dynamical systems
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
68P99 Theory of data
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.

Software:

FastICA
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bandle, C.: Isoperimetric inequalities and applications. Pitman, Monographs and studies in mathematics (1980)
[2] Baxter, JR; Rosenthal, JS, Rates of convergence for everywhere-positive Markov chains, Stat. Probab. Lett., 22, 333-338, (1995) · Zbl 0819.60056
[3] Brunton, SL; Proctor, JL; Tu, JH; Kutz, JN, Compressed sensing and dynamic mode decomposition, J. Comput. Dyn., 2, 165-191, (2015) · Zbl 1347.94012
[4] Brunton, BW; Johnson, LA; Ojemann, JG; Kutz, JN, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition, J. Neurosci. Methods, 258, 1-15, (2016)
[5] Budišić, M; Mohr, R; Mezić, I, Applied koopmanism, Appl. Chaos An Interdiscip. J. Nonlinear Sci., 22, 047510, (2012) · Zbl 1319.37013
[6] Chen, KK; Tu, JH; Rowley, CW, Variants of dynamic mode decomposition: boundary condition, koopman, and Fourier analyses, J. Nonlinear Sci., 22, 887-915, (2012) · Zbl 1259.35009
[7] Coifman, RR; Kevrekidis, IG; Lafon, S; Maggioni, M; Nadler, B, Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems, Multiscale Model. Simul., 7, 842-864, (2008) · Zbl 1175.60058
[8] Dellnitz, M; Junge, O, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36, 491-515, (1999) · Zbl 0916.58021
[9] Deuflhard, P; Weber, M, Robust Perron cluster analysis in conformation dynamics, Linear Algebra Appl., 398, 161-184, (2005) · Zbl 1070.15019
[10] Djurdjevac, N; Sarich, M; Schütte, C, Estimating the eigenvalue error of Markov state models, Multiscale Model. Simul., 10, 61-81, (2012) · Zbl 1253.65010
[11] Ferguson, AL; Panagiotopoulos, AZ; Kevrekidis, IG; Debenedetti, PG, Nonlinear dimensionality reduction in molecular simulation: the diffusion map approach, Chem. Phys. Lett., 509, 1-11, (2011)
[12] Froyland, G; Padberg, K, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238, 1507-1523, (2009) · Zbl 1178.37119
[13] Froyland, G; Padberg-Gehle, K; Bahsoun, W (ed.); Bose, CH (ed.); Froyland, G (ed.), Almost-invariant and finite-time coherent sets: directionality, duration, and diffusion, 171-216, (2014), New York · Zbl 1352.37078
[14] Froyland, G; Junge, O; Koltai, P, Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach, SIAM J. Numer. Anal., 51, 223-247, (2013) · Zbl 1267.37101
[15] Froyland, G; Gottwald, G; Hammerlindl, A, A computational method to extract macroscopic variables and their dynamics in multiscale systems, SIAM J. Appl. Dyn. Syst., 13, 1816-1846, (2014) · Zbl 1320.37012
[16] Giannakis, D.: Data-driven spectral decomposition and forecasting of ergodic dynamical systems. rXiv e-prints (2015) · Zbl 1273.62145
[17] Hopf, E, The general temporally discrete markoff process, J. Ration. Mech. Anal., 3, 13-45, (1954) · Zbl 0055.36705
[18] Hyvärinen, A., Karhunen, J., Oja, E.: Independent Component Analysis. Wiley, Hoboken (2001)
[19] Jovanović, MR; Schmid, PJ; Nichols, JW, Sparsity-promoting dynamic mode decomposition, Phys. Fluids, 26, 024103, (2014)
[20] Junge, O; Koltai, P, Discretization of the Frobenius-Perron operator using a sparse Haar tensor basis: the sparse Ulam method, SIAM J. Numer. Anal., 47, 3464-3485, (2009) · Zbl 1210.37061
[21] Klus, S., Gelß, P., Peitz, S. and Schütte, Ch.: Tensor-based dynamic mode decomposition. ArXiv e-prints (2016) · Zbl 1404.65313
[22] Klus, S. and Schütte, CH.: Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator. J. Comput. Dyn. 3(2) (2016) · Zbl 1369.37080
[23] Klus, S; Koltai, P; Schütte, Ch, On the numerical approximation of the Perron-Frobenius and koopman operator, J. Comput. Dyn., 3, 51-79, (2016) · Zbl 1353.37154
[24] Koopman, B, Hamiltonian systems and transformation in Hilbert space, Proc. Ntl. Acad. Sci. USA, 17, 315, (1931) · Zbl 0002.05701
[25] Korda, M. and Mezić, I.: On convergence of Extended Dynamic Mode Decomposition to the Koopman operator. arXiv preprint rXiv:1703.04680 (2017)
[26] Krengel, U.: Ergodic Theorems, Volume 6 of de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin (1985) · Zbl 0471.28011
[27] Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic mode decomposition: data-driven modeling of complex systems. SIAM (2016) · Zbl 1365.65009
[28] Lasota, A.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics volume 97 of Applied Mathematical Sciences, 2nd edn. Springer, Berlin (1994) · Zbl 0784.58005
[29] Leimkuhler, B., Chipot, Ch., Elber, R., Laaksonen, A., Mark, A., Schlick, T., Schütte, Ch., Skeel, R.: New Algorithms for Macromolecular Simulation (Lecture Notes in Computational Science and Engineering). Springer, New York (2006) · Zbl 1084.81003
[30] Mairal, J., Bach, F., Ponce, J., Sapiro, G.: Online dictionary learning for sparse coding. In: Proceedings of the 26th Annual International Conference on Machine Learning, ICML ’09, pp. 689-696. ACM (2009) · Zbl 1242.62087
[31] McGibbon, RT; Pande, VS, Variational cross-validation of slow dynamical modes in molecular kinetics, J. Chem. Phys., 142, 03b621, (2015)
[32] Mezić, I, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dyn., 41, 309-325, (2005) · Zbl 1098.37023
[33] Mezić, I, Analysis of fluid flows via spectral properties of the koopman operator, Ann. Rev. Fluid Mech., 45, 357-378, (2013) · Zbl 1359.76271
[34] Molgedey, L; Schuster, HG, Separation of a mixture of independent signals using time delayed correlations, Phys. Rev. Lett., 72, 3634-3637, (1994)
[35] Nadler, B; Lafon, S; Coifman, RR; Kevrekidis, IG, Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmonic Anal., 21, 113-127, (2006) · Zbl 1103.60069
[36] Noé, F; Clementi, C, Kinetic distance and kinetic maps from molecular dynamics simulation, J. Chem. Theory Comput., 11, 5002-5011, (2015)
[37] Noé, F; Nüske, F, A variational approach to modeling slow processes in stochastic dynamical systems, Multiscale Model. Simul., 11, 635-655, (2013) · Zbl 1306.65013
[38] Noé, F; Wu, H; Prinz, J-H; Plattner, N, Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules, J. Chem. Phys., 139, 184114, (2013)
[39] Nüske, F; Keller, BG; Pérez-Hernández, G; Mey, ASJS; Noé, F, Variational approach to molecular kinetics, J. Chem. Theory Comput., 10, 1739-1752, (2014)
[40] Nüske, F; Keller, BG; Pérez-Hernández, G; Mey, ASJS; Noé, F, Variational approach to molecular kinetics, J. Chem. Theory Comput., 10, 1739-1752, (2014)
[41] Nüske, F; Schneider, R; Vitalini, F; Noé, F, Variational tensor approach for approximating the rare-event kinetics of macromolecular systems, J. Chem. Phys., 144, 054105, (2016)
[42] Øksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2003) · Zbl 1025.60026
[43] Pavliotis, G.A.: Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations volume 60 of Texts in Applied Mathematics. Springer, Berlin (2014) · Zbl 1318.60003
[44] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) · Zbl 0516.47023
[45] Pérez-Hernández, G; Paul, F; Giorgino, T; Fabritiis, G; Noé, F, Identification of slow molecular order parameters for Markov model construction, J. Chem. Phys., 139, 07b604, (2013)
[46] Prinz, J-H; Wu, H; Sarich, M; Keller, B; Senne, M; Held, M; Chodera, JD; Schütte, C; Noé, F, Markov models of molecular kinetics: generation and validation, J. Chem. Phys., 134, 174105, (2011)
[47] Röblitz, S; Weber, M, Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification, Adv. Data Anal. Classif., 7, 147-179, (2013) · Zbl 1273.62145
[48] Rowley, CW; Mezić, I; Bagheri, S; Schlatter, P; Henningson, DS, Spectral analysis of nonlinear flows, J. Fluid Mech., 641, 115-127, (2009) · Zbl 1183.76833
[49] Sarich, M; Noé, F; Schütte, C, On the approximation quality of Markov state models, Multiscale Model. Simul., 8, 1154-1177, (2010) · Zbl 1210.60082
[50] Schmid, PJ, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28, (2010) · Zbl 1197.76091
[51] Schütte, Ch. and Sarich, M.: Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches. Number 24 in Courant Lecture Notes. American Mathematical Society, (2013) · Zbl 1253.65010
[52] Schütte, Ch; Fischer, A; Huisinga, W; Deuflhard, P, A direct approach to conformational dynamics based on hybrid Monte Carlo, J. Comput. Phys., 151, 146-168, (1999) · Zbl 0933.65145
[53] Schwantes, CR; Pande, VS, Improvements in Markov state model construction reveal many non-native interactions in the folding of NTL9, J. Chem. Theory. Comput., 9, 2000-2009, (2013)
[54] Shadden, SC; Lekien, F; Marsden, JE, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Phys. D Nonlinear Phenom., 212, 271-304, (2005) · Zbl 1161.76487
[55] Tong, L., Soon, V.C., Huang, Y.F., Liu, R.: AMUSE: a new blind identification algorithm. In: IEEE International Symposium on Circuits and Systems, pp. 1784-1787 (1990) · Zbl 1259.35009
[56] Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L. ,Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), (2014) · Zbl 1346.37064
[57] Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publisher, Hoboken (1960) · Zbl 0086.24101
[58] Williams, MO; Kevrekidis, IG; Rowley, CW, A data-driven approximation of the koopman operator: extending dynamic mode decomposition, J. Nonlinear Sci., 25, 1307-1346, (2015) · Zbl 1329.65310
[59] Williams, MO; Rowley, CW; Kevrekidis, IG, A kernel-based method for data-driven koopman spectral analysis, J. Comput. Dyn., 2, 247-265, (2015) · Zbl 1366.37144
[60] Wu, H; Nüske, F; Paul, F; Klus, S; Koltai, P; Noé, F, Variational koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations, J. Chem. Phys., 146, 154104, (2017)
[61] Ziehe, A. and Müller, K.-R.: TDSEP—an efficient algorithm for blind separation using time structure. In: CANN 98, pp. 675-680. Springer Science and Business Media (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.