6685
2018
eng
471
512
2
28
article
0
2017-10-12
--
--
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.
Jounal of Nonlinear Science
10.1007/s00332-017-9415-0
yes
urn:nbn:de:0297-zib-63822
Andreas Bittracher
Paulina Bressel
Péter Koltai
Stefan Klus
Ralf Banisch
Michael Dellnitz
Christof Schütte
Numerical Mathematics
Schütte, Christof
SFB-1114-B3
6670
2017
eng
article
0
--
--
--
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.
Journal of Machine Learning Research
1712.01572
no
under review
Stefan Klus
Ingmar Schuster
Ingmar Schuster
Krikamol Muandet
Numerical Mathematics
no-project
Schuster, Ingmar
6246
2016
eng
139
161
2
3
article
0
--
--
--
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.
Journal of Computational Dynamics
10.3934/jcd.2016007
yes
Stefan Klus
Erlinda Körnig
Christof Schütte
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
SFB1114-C3
MODAL-Gesamt
6267
2018
eng
985
1010
3
28
article
0
2018-01-03
--
--
Data-driven model reduction and transfer operator approximation
Journal of Nonlinear Science
https://link.springer.com/article/10.1007/s00332-017-9437-7
10.1007/s00332-017-9437-7
yes
Stefan Klus
Erlinda Koernig
Feliks Nüske
Peter Koltai
Hao Wu
Ioannis Kevrekidis
Christof Schütte
Frank Noé
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
SFB1114-A5
MODAL-Gesamt
6250
2016
eng
51
77
1
3
article
0
--
--
--
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.
Journal of Computational Dynamics
10.3934/jcd.2016003
yes
Stefan Klus
Erlinda Körnig
Peter Koltai
Christof Schütte
Numerical Mathematics
Schütte, Christof
ECMath-CH6
MODAL-MedLab
SFB1114-A5
MODAL-Gesamt
6260
2018
eng
7
31
article
IOP Publishing Ltd & London Mathematical Society
0
2018-06-04
--
--
Tensor-based dynamic mode decomposition
Nonlinearity
10.1088/1361-6544/aabc8f
yes
accepted for publication on 2018-04-09
2018-04-09
Stefan Klus
Erlinda Körnig
Patrick Gelß
Sebastian Peitz
Christof Schütte
Numerical Mathematics
Schütte, Christof
ECMATH-CH2
SFB-1114-B3
MODAL-Gesamt
6382
eng
reportzib
0
--
2017-05-03
--
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.
1438-0064
urn:nbn:de:0297-zib-63822
yes
accepted for publication on 2017-09-23
Andreas Bittracher
Paulina Bressel
Péter Koltai
Stefan Klus
Ralf Banisch
Michael Dellnitz
Christof Schütte
ZIB-Report
17-22
eng
uncontrolled
metastability
eng
uncontrolled
slow dynamics
eng
uncontrolled
effective dynamics
eng
uncontrolled
transition manifold
eng
uncontrolled
embedding
eng
uncontrolled
transfer operator
eng
uncontrolled
reaction coordinate
Operators on function spaces (general)
Computational methods for stochastic equations [See also 65C30]
Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10]
Numerical Mathematics
Schütte, Christof
SFB1114-C3
https://opus4.kobv.de/opus4-zib/files/6382/ZIB-Report_17-22.pdf
7317
2019
eng
6
14
article
0
2019-04-08
--
--
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.
Journal of Computational and Nonlinear Dynamics
10.1115/1.4043148
yes
Patrick Gelß
Erlinda Körnig
Stefan Klus
Jens Eisert
Christof Schütte
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
MODAL-Gesamt
7341
2019
eng
2000
2046
4
18
article
0
--
--
--
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.
SIAM Journal on Applied Dynamical Systems (SIADS)
1902.04920
yes
Wei Zhang
Wei Zhang
Stefan Klus
Tim Conrad
Christof Schütte
Numerical Mathematics
Visual Data Analysis
Schütte, Christof
Conrad, Tim
MODAL-MedLab
MODAL-Gesamt
Zhang, Wei
7773
2019
eng
12
29
article
0
--
--
--
Kernel methods for detecting coherent structures in dynamical data
Chaos: An Interdisciplinary Journal of Nonlinear Science
10.1063/1.5100267
yes
Stefan Klus
Erlinda Körnig
Brooke E. Husic
Mattes Mollenhauer
Frank Noe
Numerical Mathematics
MathPlus - AA1-1
Mollenhauer, Mattes
7107
2018
eng
24
149
article
0
--
2018-12-28
--
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.
Journal of Chemical Physics
10.1063/1.5063533
yes
2018-12-04
Stefan Klus
Erlinda Körnig
Andreas Bittracher
Ingmar Schuster
Christof Schütte
Numerical Mathematics
Schütte, Christof
ECMATH-CH2
SFB-1114-B3
7588
2019
eng
article
0
--
--
--
Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds
Journal of Nonlinear Science
yes
accepted for publication on 2020-08-15
Andreas Bittracher
Erlinda Körnig
Stefan Klus
Boumediene Hamzi
Christof Schütte
Numerical Mathematics
Schütte, Christof
MathPlus - AA1-1
8030
2020
eng
96
5
article
0
--
--
--
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.
Applied Network Science
10.1007/s41109-020-00339-2
yes
Kateryna Melnyk
Kalina Tsankova
Grègoire Montavon
Stefan Klus
Tim Conrad
Visual Data Analysis
Conrad, Tim
MODAL-MedLab
MODAL-Gesamt
Visual and Data-centric Computing
7589
2020
eng
406
article
0
2020-03-06
--
--
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
Physica D
10.1016/j.physd.2020.132416
yes
epub ahead of print
Stefan Klus
Erlinda Körnig
Feliks Nüske
Sebastian Peitz
Jan-Hendrik Niemann
Cecilia Clementi
Christof Schütte
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
MODAL-Gesamt
Niemann, Jan-Hendrik
MathPlus - AA1-1
Modeling and Simulation of Complex Processes
7791
2020
eng
109
131
304
incollection
Springer International
0
--
--
--
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Advances om Dynamics, Optimization and Computation.
Series: Studies in Systems, Decision and Control. A volume dedicated to Michael Dellnitz on his 60th birthday
978-3-030-51264-4
yes
Mattes Mollenhauer
Erlinda Körnig
Oliver Junge
Ingmar Schuster
O. Schütze
Stefan Klus
Gary Froyland
Christof Schütte
S. Ober-Blobaum
K. Padberg-Gehle
Numerical Mathematics
Schütte, Christof
MODAL-MedLab
MODAL-Gesamt
MathPlus - AA1-1
Mollenhauer, Mattes
Modeling and Simulation of Complex Processes
8186
2021
eng
article
0
--
--
--
Feature space approximation for kernel-based supervised learning
Knowledge-Based Sytems
2011.12651v1
yes
accepted for publication
2021-03-04
publish
Patrick Gelss
Erlinda Körnig
Stefan Klus
Ingmar Schuster
Christof Schütte
Schütte, Christof
ECMATH-CH2
MODAL-MedLab
SFB-1114-B3
MODAL-Gesamt
Modeling and Simulation of Complex Processes
8106
2021
eng
5
16
article
0
--
2021-05-13
--
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.
PLOS ONE
2012.07718
10.1371/journal.pone.0250970
yes
2021-04-19
Jan-Hendrik Niemann
Jan-Hendrik Niemann
Stefan Klus
Christof Schütte
Numerical Mathematics
Schütte, Christof
Niemann, Jan-Hendrik
MathPlus-EF4-3
Modeling and Simulation of Complex Processes