Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen
OPUS4-6685 Wissenschaftlicher Artikel Bittracher, Andreas; Koltai, Péter; Klus, Stefan; Banisch, Ralf; Dellnitz, Michael; Schütte, Christof Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics 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. 2018 41 Jounal of Nonlinear Science 28 2 471 512 10.1007/s00332-017-9415-0 Numerical Mathematics
OPUS4-6670 Wissenschaftlicher Artikel Klus, Stefan; Schuster, Ingmar; Muandet, Krikamol Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces 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. 2017 Journal of Machine Learning Research Numerical Mathematics
OPUS4-6246 Wissenschaftlicher Artikel Klus, Stefan; Schütte, Christof Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator 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. 2016 22 Journal of Computational Dynamics 3 2 139 161 10.3934/jcd.2016007 Numerical Mathematics
OPUS4-6267 Wissenschaftlicher Artikel Klus, Stefan; Nüske, Feliks; Koltai, Peter; Wu, Hao; Kevrekidis, Ioannis; Schütte, Christof; Noé, Frank Data-driven model reduction and transfer operator approximation 2018 25 Journal of Nonlinear Science 28 3 985 1010 10.1007/s00332-017-9437-7 Numerical Mathematics
OPUS4-6250 Wissenschaftlicher Artikel Klus, Stefan; Koltai, Peter; Schütte, Christof On the numerical approximation of the Perron-Frobenius and Koopman operator 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. 2016 26 Journal of Computational Dynamics 3 1 51 77 10.3934/jcd.2016003 Numerical Mathematics
OPUS4-6260 Wissenschaftlicher Artikel Klus, Stefan; Gelß, Patrick; Peitz, Sebastian; Schütte, Christof Tensor-based dynamic mode decomposition IOP Publishing Ltd & London Mathematical Society 2018 Nonlinearity 31 7 10.1088/1361-6544/aabc8f Numerical Mathematics
OPUS4-6382 misc Bittracher, Andreas; Koltai, Péter; Klus, Stefan; Banisch, Ralf; Dellnitz, Michael; Schütte, Christof Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics 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. 2017 urn:nbn:de:0297-zib-63822 Numerical Mathematics
OPUS4-7317 Wissenschaftlicher Artikel Gelß, Patrick; Klus, Stefan; Eisert, Jens; Schütte, Christof Multidimensional Approximation of Nonlinear Dynamical Systems 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. 2019 Journal of Computational and Nonlinear Dynamics 14 6 10.1115/1.4043148 Numerical Mathematics
OPUS4-7341 Wissenschaftlicher Artikel Zhang, Wei; Klus, Stefan; Conrad, Tim; Schütte, Christof Learning chemical reaction networks from trajectory data We develop a data-driven method to learn chemical reaction networks from trajectory data. Modeling the reaction system as a continuous-time Markov chain and assuming the system is fully observed,our method learns the propensity functions of the system with predetermined basis functions by maximizing the likelihood function of the trajectory data under l^1 sparse regularization. We demonstrate our method with numerical examples using synthetic data and carry out an asymptotic analysis of the proposed learning procedure in the infinite-data limit. 2019 46 SIAM Journal on Applied Dynamical Systems (SIADS) 18 4 2000 2046 Numerical Mathematics
OPUS4-7773 Wissenschaftlicher Artikel Klus, Stefan; Husic, Brooke E.; Mollenhauer, Mattes; Noe, Frank Kernel methods for detecting coherent structures in dynamical data 2019 Chaos: An Interdisciplinary Journal of Nonlinear Science 29 12 10.1063/1.5100267 Numerical Mathematics
OPUS4-7107 Wissenschaftlicher Artikel Klus, Stefan; Bittracher, Andreas; Schuster, Ingmar; Schütte, Christof A kernel-based approach to molecular conformation analysis 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. 2018 Journal of Chemical Physics 149 24 10.1063/1.5063533 Numerical Mathematics
OPUS4-7588 Wissenschaftlicher Artikel Bittracher, Andreas; Klus, Stefan; Hamzi, Boumediene; Schütte, Christof Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds 2019 Journal of Nonlinear Science Numerical Mathematics
OPUS4-8030 Wissenschaftlicher Artikel Melnyk, Kateryna; Montavon, Grègoire; Klus, Stefan; Conrad, Tim Graph Kernel Koopman Embedding for Human Microbiome Analysis More and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, or some cancer types. Thanks to modern high-throughput omics technologies, it becomes possible to directly analyze human microbiome and its influence on the health status. Microbial communities are monitored over long periods of time and the associations between their members are explored. These relationships can be described by a time-evolving graph. In order to understand responses of the microbial community members to a distinct range of perturbations such as antibiotics exposure or diseases and general dynamical properties, the time-evolving graph of the human microbial communities has to be analyzed. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to solving this problem is the representation of the time-evolving graphs as fixed-length feature vectors preserving the original dynamics. We propose a method for learning the embedding of the time-evolving graph that is based on the spectral analysis of transfer operators and graph kernels. We demonstrate that our method can capture temporary changes in the time-evolving graph on both synthetic data and real-world data. Our experiments demonstrate the efficacy of the method. Furthermore, we show that our method can be applied to human microbiome data to study dynamic processes. 2020 Applied Network Science 5 96 10.1007/s41109-020-00339-2 Visual Data Analysis
OPUS4-7589 Wissenschaftlicher Artikel Klus, Stefan; Nüske, Feliks; Peitz, Sebastian; Niemann, Jan-Hendrik; Clementi, Cecilia; Schütte, Christof Data-driven approximation of the Koopman generator: Model reduction, system identification, and control 2020 Physica D 406 10.1016/j.physd.2020.132416 Numerical Mathematics
OPUS4-7791 misc Mollenhauer, Mattes; Schuster, Ingmar; Klus, Stefan; Schütte, Christof Junge, Oliver; Schütze, O.; Froyland, Gary; Ober-Blobaum, S.; Padberg-Gehle, K. Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces Springer International 2020 22 Advances om Dynamics, Optimization and Computation.
Series: Studies in Systems, Decision and Control. A volume dedicated to Michael Dellnitz on his 60th birthday 304 978-3-030-51264-4 109 131 Numerical Mathematics
OPUS4-8186 Wissenschaftlicher Artikel Gelss, Patrick; Klus, Stefan; Schuster, Ingmar; Schütte, Christof Feature space approximation for kernel-based supervised learning 2021 Knowledge-Based Sytems Modeling and Simulation of Complex Processes
OPUS4-8106 Wissenschaftlicher Artikel Niemann, Jan-Hendrik; Klus, Stefan; Schütte, Christof Data-driven model reduction of agent-based systems using the Koopman generator 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. 2021 PLOS ONE 16 5 10.1371/journal.pone.0250970 Numerical Mathematics