@article{NiemannWinkelmannWolfetal.2021, author = {Niemann, Jan-Hendrik and Winkelmann, Stefanie and Wolf, Sarah and Sch{\"u}tte, Christof}, title = {Agent-based modeling: Population limits and large timescales}, volume = {31}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, number = {3}, issn = {1438-0064}, doi = {10.1063/5.0031373}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-77309}, year = {2021}, abstract = {Modeling, simulation and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from informal descriptions of interaction dynamics to more formal, mathematical models. In this paper, we study agent-based models (ABMs) given as continuous-time stochastic processes and their pathwise approximation by ordinary and stochastic differential equations (ODEs and SDEs, respectively) for medium to large populations. By means of an appropriately adapted transfer operator approach we study the behavior of the ABM process on long time scales. We show that, under certain conditions, the transfer operator approach allows to bridge the gap between the pathwise results for large populations on finite timescales, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales like large deviation theory. The latter provides a rigorous analysis of rare events including the associated asymptotic rates on timescales that scale exponentially with the population size. We demonstrate that it is possible to reveal metastable structures and timescales of rare events of the ABM process by finite-length trajectories of the SDE process for large enough populations. This approach has the potential to drastically reduce computational effort for the analysis of ABMs.}, language = {en} } @article{NiemannKlusSchuette2021, author = {Niemann, Jan-Hendrik and Klus, Stefan and Sch{\"u}tte, Christof}, title = {Data-driven model reduction of agent-based systems using the Koopman generator}, volume = {16}, journal = {PLOS ONE}, number = {5}, arxiv = {http://arxiv.org/abs/2012.07718}, doi = {10.1371/journal.pone.0250970}, year = {2021}, abstract = {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.}, language = {en} } @misc{NiemannSchuetteKlus2021, author = {Niemann, Jan-Hendrik and Sch{\"u}tte, Christof and Klus, Stefan}, title = {Simulation data: Data-driven model reduction of agent-based systems using the Koopman generator}, volume = {16}, journal = {PLOS ONE}, number = {5}, doi = {http://doi.org/10.5281/zenodo.4522119}, year = {2021}, language = {en} } @phdthesis{Niemann2022, author = {Niemann, Jan-Hendrik}, title = {Learning Reduced Models for Large-Scale Agent-Based Systems}, doi = {10.17169/refubium-35245}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:188-refubium-35530-6}, pages = {130}, year = {2022}, abstract = {Modeling social systems and studying their dynamical behavior plays an important role in many fields of research. Agent-based modeling provides a high degree of detail into artificial societies by describing the model from the perspective of the agents. The interactions of agents, often characterized by simple rules, lead to complex, time-evolving patterns. Their understanding is of great importance, e.g., for predicting and influencing epidemics. Analysis and simulation, however, often becomes prohibitively time-consuming when the number of agents or the time scale of interest is large. Therefore, this thesis is devoted to learn significantly reduced models of large-scale agent-based systems from simulation data. We show how data-driven methods based on transfer operators can be used to find reduced models represented by ordinary or stochastic differential equations that describe the dynamical behavior of larger groups or entire populations and thus enable the analysis and prediction of agent-based systems. To this end, we first present an extension of EDMD (extended dynamic mode decomposition) called gEDMD to approximate the Koopman generator from data. This method can be used to compute eigenfunctions, eigenvalues, and modes of the generator, as well as for system identification and model reduction of both deterministic and non-deterministic dynamical systems. Secondly, we analyze the long-term behavior of certain agent-based models and their pathwise approximations by stochastic differential equations for large numbers of agents using transfer operators. We show that, under certain conditions, the transfer operator approach connects the pathwise approximations on finite time scales with methods for describing the behavior on possibly exponentially long time scales. As a consequence, we can use the finite-time, pathwise approximations to characterize metastable behavior on long time scales using transfer operators. This can significantly reduce the computational cost. The third part addresses the data-driven model reduction since in many cases no analytical limit models are known or existent. We show how the Koopman operator theory can be used to infer the governing equations of agent-based systems directly from simulation data. Using benchmark problems, we demonstrate that for sufficiently large population sizes the data-driven models agree well with analytical limit equations and, moreover, that the reduced models allow predictions even in cases far from the limit or when no limit equations are known. Lastly, we demonstrate the potential of the presented approach. We present an ansatz for the multi-objective optimization of agent-based systems with the help of data-driven surrogate models based on the Koopman generator. In particular, when limit models are unknown or non-existent, this approach makes multi-objective optimization problems solvable that would otherwise be computationally infeasible due to very expensive objective functions.}, language = {en} }