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 - GEN 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 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. T3 - ZIB-Report - 17-22 KW - metastability KW - slow dynamics KW - effective dynamics KW - transition manifold KW - embedding KW - transfer operator KW - reaction coordinate Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-63822 SN - 1438-0064 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 - Gelß, Patrick A1 - Peitz, Sebastian A1 - Schütte, Christof T1 - Tensor-based dynamic mode decomposition JF - Nonlinearity Y1 - 2018 U6 - https://doi.org/10.1088/1361-6544/aabc8f VL - 31 IS - 7 PB - IOP Publishing Ltd & London Mathematical Society ER - TY - GEN A1 - Mollenhauer, Mattes A1 - Schuster, Ingmar A1 - Klus, Stefan A1 - Schütte, Christof ED - Junge, Oliver ED - Schütze, O. ED - Froyland, Gary ED - Ober-Blobaum, S. ED - Padberg-Gehle, K. T1 - Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces T2 - Advances om Dynamics, Optimization and Computation. Series: Studies in Systems, Decision and Control. A volume dedicated to Michael Dellnitz on his 60th birthday Y1 - 2020 SN - 978-3-030-51264-4 U6 - https://doi.org/10.1007/978-3-030-51264-4_5 VL - 304 SP - 109 EP - 131 PB - Springer International ER - TY - GEN A1 - Niemann, Jan-Hendrik A1 - Schütte, Christof A1 - Klus, Stefan T1 - Simulation data: Data-driven model reduction of agent-based systems using the Koopman generator T2 - PLOS ONE Y1 - 2021 U6 - https://doi.org/http://doi.org/10.5281/zenodo.4522119 N1 - This repository contains the simulation data for the article "Data-driven model reduction of agent-based systems using the Koopman generator" by Jan-Hendrik Niemann, Stefan Klus and Christof Schütte. The archive complete_voter_model.zip contains the simulation results for the extended voter model on a complete graph for the parameters given in the corresponding txt-files to learn a reduced SDE model. The files are of the form [types, time steps, samples, training points].The archive dependency.zip contains additional simulation results of the form [types, time steps, samples, training points] to learn a reduced SDE model. The parameters used are given in the corresponding txt-files.The archive random_voter_model.zip contains the simulation results to learn a reduced SDE model for the given adjacency matrix within the archive. The file aggregate_state is of the form [training points, types, time steps, samples]. The file full_state is of the form [training points, agents, time steps, samples].The archive predator_prey_model.zip contains the simulation results to learn a reduced SDE model and calculation of the mean value of the agent-based model. The data is of the form [types, time steps, samples, training points] and [samples, time steps, types].The archive two_clustered_voter_model.zip contains the simulation results for the extended voter model on a graph with two clusters for the given adjacency matrices to learn a reduced SDE model. The file aggregate_state is of the form [training points, types, time steps, samples]. The file full_state is of the form [training points, agents, time steps, samples]. VL - 16 IS - 5 ER - TY - GEN A1 - Schütte, Christof A1 - Klus, Stefan A1 - Hartmann, Carsten T1 - Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning N2 - One of the main challenges in molecular dynamics is overcoming the “timescale barrier”, a phrase used to describe that in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, not even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics, and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behavior on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory as well as the algorithmic development from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in MD. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject. T3 - ZIB-Report - 22-25 Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-88637 SN - 1438-0064 ER - TY - JOUR A1 - Schütte, Christof A1 - Klus, Stefan A1 - Hartmann, Carsten T1 - Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning JF - Acta Numerica N2 - One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject. Y1 - 2023 U6 - https://doi.org/10.1017/S0962492923000016 VL - 32 SP - 517 EP - 673 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 - 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 -