TY - JOUR A1 - Niemann, Jan-Hendrik A1 - Klus, Stefan A1 - Conrad, Natasa Djurdjevac A1 - Schütte, Christof T1 - Koopman-Based Surrogate Models for Multi-Objective Optimization of Agent-Based Systems JF - Physica D: Nonlinear Phenomena N2 - Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend on potentially nonlinear control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems. Y1 - 2024 U6 - https://doi.org/https://doi.org/10.1016/j.physd.2024.134052 VL - 460 SP - 134052 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 - 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 - Gelß, Patrick A1 - Klus, Stefan A1 - Knebel, Sebastian A1 - Shakibaei, Zarin A1 - Pokutta, Sebastian T1 - Low-Rank Tensor Decompositions of Quantum Circuits JF - Journal of Computational Physics N2 - Quantum computing is arguably one of the most revolutionary and disruptive technologies of this century. Due to the ever-increasing number of potential applications as well as the continuing rise in complexity, the development, simulation, optimization, and physical realization of quantum circuits is of utmost importance for designing novel algorithms. We show how matrix product states (MPSs) and matrix product operators (MPOs) can be used to express certain quantum states, quantum gates, and entire quantum circuits as low-rank tensors. This enables the analysis and simulation of complex quantum circuits on classical computers and to gain insight into the underlying structure of the system. We present different examples to demonstrate the advantages of MPO formulations and show that they are more efficient than conventional techniques if the bond dimensions of the wave function representation can be kept small throughout the simulation. Y1 - 2022 ER - TY - JOUR A1 - Klus, Stefan A1 - Djurdjevac Conrad, Natasa T1 - Koopman-based spectral clustering of directed and time-evolving graphs JF - Journal of Nonlinear Science N2 - While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows. Y1 - 2022 U6 - https://doi.org/10.1007/s00332-022-09863-0 VL - 33 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 - Gelss, Patrick A1 - Klus, Stefan A1 - Schuster, Ingmar A1 - Schütte, Christof T1 - Feature space approximation for kernel-based supervised learning JF - Knowledge-Based Sytems Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1016/j.knosys.2021.106935 VL - 221 PB - Elsevier 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 - 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 - 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 -