TY - JOUR A1 - Gelß, Patrick A1 - Klus, Stefan A1 - Eisert, Jens A1 - Schütte, Christof T1 - Multidimensional Approximation of Nonlinear Dynamical Systems JF - Journal of Computational and Nonlinear Dynamics N2 - A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems. Y1 - 2019 U6 - https://doi.org/10.1115/1.4043148 VL - 14 IS - 6 ER - TY - JOUR A1 - Klus, Stefan A1 - Bittracher, Andreas A1 - Schuster, Ingmar A1 - Schütte, Christof T1 - A kernel-based approach to molecular conformation analysis JF - Journal of Chemical Physics N2 - We present a novel machine learning approach to understanding conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov State Models, EDMD, and TICA can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular the alanine dipeptide and the protein NTL9. Y1 - 2018 U6 - https://doi.org/10.1063/1.5063533 VL - 149 IS - 24 ER - TY - JOUR A1 - Bittracher, Andreas A1 - Klus, Stefan A1 - Hamzi, Boumediene A1 - Schütte, Christof T1 - Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds JF - Journal of Nonlinear Science N2 - We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches. Y1 - 2021 U6 - https://doi.org/10.1007/s00332-020-09668-z VL - 31 ER - TY - JOUR A1 - Klus, Stefan A1 - Nüske, Feliks A1 - Peitz, Sebastian A1 - Niemann, Jan-Hendrik A1 - Clementi, Cecilia A1 - Schütte, Christof T1 - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control JF - Physica D: Nonlinear Phenomena Y1 - 2020 U6 - https://doi.org/10.1016/j.physd.2020.132416 VL - 406 ER - TY - JOUR A1 - Klus, Stefan A1 - Husic, Brooke E. A1 - Mollenhauer, Mattes A1 - Noe, Frank T1 - Kernel methods for detecting coherent structures in dynamical data JF - Chaos: An Interdisciplinary Journal of Nonlinear Science Y1 - 2019 U6 - https://doi.org/10.1063/1.5100267 VL - 29 IS - 12 ER - TY - JOUR A1 - Bittracher, Andreas A1 - Koltai, Péter A1 - Klus, Stefan A1 - Banisch, Ralf A1 - Dellnitz, Michael A1 - Schütte, Christof T1 - Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics JF - Jounal of Nonlinear Science N2 - We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics. Y1 - 2018 U6 - https://doi.org/10.1007/s00332-017-9415-0 VL - 28 IS - 2 SP - 471 EP - 512 ER - TY - JOUR A1 - Klus, Stefan A1 - Schuster, Ingmar A1 - Muandet, Krikamol T1 - Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces JF - Journal of Machine Learning Research N2 - Transfer operators such as the Perron-Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. In fact, most of the existing methods can be derived from our framework, providing a unifying view on the approximation of transfer operators. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis. Y1 - 2017 ER - TY - JOUR A1 - Niemann, Jan-Hendrik A1 - Klus, Stefan A1 - Schütte, Christof T1 - Data-driven model reduction of agent-based systems using the Koopman generator JF - PLOS ONE N2 - The dynamical behavior of social systems can be described by agent-based models. Although single agents follow easily explainable rules, complex time-evolving patterns emerge due to their interaction. The simulation and analysis of such agent-based models, however, is often prohibitively time-consuming if the number of agents is large. In this paper, we show how Koopman operator theory can be used to derive reduced models of agent-based systems using only simulation or real-world data. Our goal is to learn coarse-grained models and to represent the reduced dynamics by ordinary or stochastic differential equations. The new variables are, for instance, aggregated state variables of the agent-based model, modeling the collective behavior of larger groups or the entire population. Using benchmark problems with known coarse-grained models, we demonstrate that the obtained reduced systems are in good agreement with the analytical results, provided that the numbers of agents is sufficiently large. Y1 - 2021 U6 - https://doi.org/10.1371/journal.pone.0250970 VL - 16 IS - 5 ER - TY - JOUR A1 - Klus, Stefan A1 - Schütte, Christof T1 - Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator JF - Journal of Computational Dynamics N2 - The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the system's behavior are the Perron-Frobenius operator and the Koopman operator. Due to the curse of dimensionality, computing the eigenfunctions of high-dimensional systems is in general infeasible. We will propose a tensor-based reformulation of two numerical methods for computing finite-dimensional approximations of the aforementioned infinite-dimensional operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD). The aim of the tensor formulation is to approximate the eigenfunctions by low-rank tensors, potentially resulting in a significant reduction of the time and memory required to solve the resulting eigenvalue problems, provided that such a low-rank tensor decomposition exists. Typically, not all variables of a high-dimensional dynamical system contribute equally to the system's behavior, often the dynamics can be decomposed into slow and fast processes, which is also reflected in the eigenfunctions. Thus, the weak coupling between different variables might be approximated by low-rank tensor cores. We will illustrate the efficiency of the tensor-based formulation of Ulam's method and EDMD using simple stochastic differential equations. Y1 - 2016 U6 - https://doi.org/10.3934/jcd.2016007 VL - 3 IS - 2 SP - 139 EP - 161 ER - TY - JOUR A1 - Klus, Stefan A1 - Koltai, Peter A1 - Schütte, Christof T1 - On the numerical approximation of the Perron-Frobenius and Koopman operator JF - Journal of Computational Dynamics N2 - Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di� fferent methods that have been developed over the last decades to compute � infinite-dimensional approximations of these in� finite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di� fferences between these approaches. The results will be illustrated using simple stochastic di� fferential equations and molecular dynamics examples. Y1 - 2016 U6 - https://doi.org/10.3934/jcd.2016003 VL - 3 IS - 1 SP - 51 EP - 77 ER -