TY - JOUR A1 - von Lindheim, Johannes A1 - Harikrishnan, Abhishek A1 - Dörffel, Tom A1 - Klein, Rupert A1 - Koltai, Peter A1 - Mikula, Natalia A1 - Müller, Annette A1 - Névir, Peter A1 - Pacey, George A1 - Polzin, Robert A1 - Vercauteren, Nikki T1 - Definition, detection and tracking of persistent structures in atmospheric flows JF - arXiv N2 - Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" - to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence. Y1 - 2021 ER - TY - JOUR A1 - Fackeldey, Konstantin A1 - Koltai, Peter A1 - Nevir, Peter A1 - Rust, Henning A1 - Schild, Axel A1 - Weber, Marcus T1 - From metastable to coherent sets - Time-discretization schemes JF - Chaos: An Interdisciplinary Journal of Nonlinear Science N2 - In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application. Y1 - 2019 U6 - https://doi.org/10.1063/1.5058128 VL - 29 SP - 012101 EP - 012101 ER - TY - GEN A1 - Fackeldey, Konstantin A1 - Koltai, Péter A1 - Névir, Peter A1 - Rust, Henning A1 - Schild, Axel A1 - Weber, Marcus T1 - From Metastable to Coherent Sets - time-discretization schemes N2 - Given a time-dependent stochastic process with trajectories x(t) in a space $\Omega$, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space $M\subset\Omega$, coherent sets $M(t)\subset\Omega$ are defined in space and time. Hence, if we extend the space by the time-variable t, coherent sets are metastable sets in $\Omega\times[0,\infty]$. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application. T3 - ZIB-Report - 17-74 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66074 SN - 1438-0064 ER - TY - GEN A1 - Koltai, Peter A1 - Ciccotti, Giovanni A1 - Schütte, Christof T1 - On metastability and Markov state models for non-stationary molecular dynamics BT - 2016 Editor's Choice of The Journal of Chemical Physics T2 - The Journal of Chemical Physics N2 - We utilize the theory of coherent sets to build Markov state models for non- equilibrium molecular dynamical systems. Unlike for systems in equilibrium, “meta- stable” sets in the non-equilibrium case may move as time evolves. We formalize this concept by relying on the theory of coherent sets, based on this we derive finite-time non-stationary Markov state models, and illustrate the concept and its main differences to equilibrium Markov state modeling on simple, one-dimensional examples. T3 - ZIB-Report - 16-11 KW - coherent set, KW - Markov state model KW - non-equilibrium molecular dynamics KW - metastability Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57869 SN - 1438-0064 VL - 174103 ET - 145 ER - TY - JOUR A1 - Koltai, Peter A1 - Ciccotti, Giovanni A1 - Schütte, Christof T1 - On Markov state models for non-equilibrium molecular dynamics JF - The Journal of Chemical Physics Y1 - 2016 U6 - https://doi.org/10.1063/1.4966157 N1 - 2016 Editor's Choice of The Journal of Chemical Physics VL - 145 IS - 174103 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 - Koltai, Peter A1 - Wu, Hao A1 - Noé, Frank A1 - Schütte, Christof T1 - Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics JF - Computation Y1 - 2018 U6 - https://doi.org/10.3390/computation6010022 VL - 6 IS - 1 PB - MDPI CY - Basel, Switzerland 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 - Klus, Stefan A1 - Nüske, Feliks A1 - Koltai, Peter A1 - Wu, Hao A1 - Kevrekidis, Ioannis A1 - Schütte, Christof A1 - Noé, Frank T1 - Data-driven model reduction and transfer operator approximation JF - Journal of Nonlinear Science Y1 - 2018 UR - https://link.springer.com/article/10.1007/s00332-017-9437-7 U6 - https://doi.org/10.1007/s00332-017-9437-7 VL - 28 IS - 3 SP - 985 EP - 1010 ER - TY - JOUR A1 - Miron, Philippe A1 - Beron-Vera, Francisco A1 - Helfmann, Luzie A1 - Koltai, Péter T1 - Transition paths of marine debris and the stability of the garbage patches JF - Chaos: An Interdisciplinary Journal of Nonlinear Science N2 - We used transition path theory (TPT) to infer "reactive" pathways of floating marine debris trajectories. The TPT analysis was applied on a pollution-aware time-homogeneous Markov chain model constructed from trajectories produced by satellite-tracked undrogued buoys from the NOAA Global Drifter Program. The latter involved coping with the openness of the system in physical space, which further required an adaptation of the standard TPT setting. Directly connecting pollution sources along coastlines with garbage patches of varied strengths, the unveiled reactive pollution routes represent alternative targets for ocean cleanup efforts. Among our specific findings we highlight: constraining a highly probable pollution source for the Great Pacific Garbage Patch; characterizing the weakness of the Indian Ocean gyre as a trap for plastic waste; and unveiling a tendency of the subtropical gyres to export garbage toward the coastlines rather than to other gyres in the event of anomalously intense winds. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1063/5.0030535 VL - 31 IS - 3 ER - TY - JOUR A1 - Koltai, Péter A1 - Schütte, Christof T1 - A multiscale perturbation expansion approach for Markov state modeling of non-stationary molecular dynamics JF - SIAM J. Multiscale Model. Simul. N2 - We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one. Y1 - 2018 U6 - https://doi.org/10.1137/17M1146403 VL - 16 IS - 4 SP - 1455 EP - 1485 PB - SIAM ER - TY - JOUR A1 - Koltai, Péter A1 - Lie, Han Cheng A1 - Plonka, Martin T1 - Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics JF - Nonlinearity N2 - We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators. Y1 - 2019 U6 - https://doi.org/10.1088/1361-6544/ab1f2a VL - 32 IS - 11 SP - 4232 EP - 4257 ER - TY - JOUR A1 - Mollenhauer, Mattes A1 - Klus, Stefan A1 - Schütte, Christof A1 - Koltai, Péter T1 - Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence JF - Journal of Machine Learning Research N2 - We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition. Y1 - 2022 UR - https://jmlr.org/papers/v23/20-442.html VL - 23 IS - 327 SP - 1 EP - 34 ER - TY - JOUR A1 - Bittracher, Andreas A1 - Mollenhauer, Mattes A1 - Koltai, Péter A1 - Schütte, Christof T1 - Optimal Reaction Coordinates: Variational Characterization and Sparse Computation JF - Multiscale Modelling & Simulation N2 - Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques. Y1 - 2023 U6 - https://doi.org/10.1137/21M1448367 VL - 21 IS - 2 SP - 449 EP - 488 ER - TY - JOUR A1 - Wulkow, Niklas A1 - Koltai, Péter A1 - Sunkara, Vikram A1 - Schütte, Christof T1 - Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory JF - J. Stat. Phys. N2 - We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples. Y1 - 2021 ER - TY - JOUR A1 - Wulkow, Niklas A1 - Koltai, Péter A1 - Schütte, Christof T1 - Memory-Based Reduced Modelling and Data-Based Estimation of Opinion Spreading JF - Journal of Nonlinear Science N2 - We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori–Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies. Y1 - 2021 U6 - https://doi.org/10.1007/s00332-020-09673-2 VL - 31 ER - TY - JOUR A1 - Helfmann, Luzie A1 - Heitzig, Jobst A1 - Koltai, Péter A1 - Kurths, Jürgen A1 - Schütte, Christof T1 - Statistical analysis of tipping pathways in agent-based models JF - Eur. Phys. J. Spec. Top. N2 - Agent-based models are a natural choice for modeling complex social systems. In such models simple stochastic interaction rules for a large population of individuals on the microscopic scale can lead to emergent dynamics on the macroscopic scale, for instance a sudden shift of majority opinion or behavior. Here we are introducing a methodology for studying noise-induced tipping between relevant subsets of the agent state space representing characteristic configurations. Due to a large number of interacting individuals, agent-based models are high-dimensional, though usually a lower-dimensional structure of the emerging collective behaviour exists. We therefore apply Diffusion Maps, a non-linear dimension reduction technique, to reveal the intrinsic low-dimensional structure. We characterize the tipping behaviour by means of Transition Path Theory, which helps gaining a statistical understanding of the tipping paths such as their distribution, flux and rate. By systematically studying two agent-based models that exhibit a multitude of tipping pathways and cascading effects, we illustrate the practicability of our approach. Y1 - 2021 U6 - https://doi.org/10.1140/epjs/s11734-021-00191-0 VL - 230 SP - 3249 EP - 3271 ER - TY - JOUR A1 - Helfmann, Luzie A1 - Ribera Borrell, Enric A1 - Schütte, Christof A1 - Koltai, Peter T1 - Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics JF - Journal of Nonlinear Science Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1007/s00332-020-09652-7 VL - 30 SP - 3321 EP - 3366 ER - TY - JOUR A1 - Lücke, Marvin A1 - Heitzig, Jobst A1 - Koltai, Péter A1 - Molkethin, Nora A1 - Winkelmann, Stefanie T1 - Large population limits of Markov processes on random networks JF - Stochastic Processes and their Applications N2 - We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called "voter model" on Erdős-Rényi random graphs, on the stochastic block model, as well as on random regular graphs. Moreover, a heterogeneous population of agents is studied. For each of these types of interaction networks, we specify the convergence conditions in dependency on the corresponding model parameters. Y1 - 2023 U6 - https://doi.org/10.1016/j.spa.2023.09.007 VL - 166 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 - Lücke, Marvin A1 - Winkelmann, Stefanie A1 - Heitzig, Jobst A1 - Molkenthin, Nora A1 - Koltai, Péter T1 - Learning interpretable collective variables for spreading processes on networks JF - Physical Review E N2 - Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically. Y1 - 2024 U6 - https://doi.org/10.1103/PhysRevE.109.L022301 VL - 2 SP - L022301 ET - 109 ER - TY - GEN A1 - Koltai, Péter A1 - Schütte, Christof T1 - A multi scale perturbation expansion approach for Markov state modeling of non-stationary molecular dynamics N2 - We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one. T3 - ZIB-Report - 17-49 KW - Markov state model KW - non-equilibrium molecular dynamics KW - two timescale master equation KW - non-stationary forcing KW - metastability Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64868 SN - 1438-0064 ER -