**Project Heads**

*Carlo Jaeger, Stefan Klus, Christof Schütte, Sarah Wolf*

**Project Members**

Jan-Hendrik Niemann (ZIB)

**Project Duration**

01.01.2019 – 31.12.2021

**Located at**

ZIB

Agent-based models are easily explainable and powerful tools to simulate complex dynamical behavior emerging in social systems. The simulation of such models however, is often being time-consuming and hard to analyze due to their high-dimensionality. The Global Climate Forum in Berlin developed the *Mobility Transition Model* (MoTMo) to support model-stakeholder interactions and decision making. MoTMo is a large-scale stochastic agent-based model which simulates the evolution of private mobility demand of a synthetic population of up to 10 million agents. Linked and influenced in a friendship network each agent maximizes its objectives such as costs, environmental impact and innovation ambitions by its mobility option. The aim of this project is to obtain simpler, coarse-grained representations from black box data of the real dynamical systems of a large-scale agent-based models. The reduced models help with validation, parameter optimization, sensitivity analysis and more.

We developed the so-called *generator extended dynamic mode decomposition* (gEDMD) to approximate the generator of the stochastic Koopman (transfer) operator. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator as well as for system identification and coarse-graining dynamical systems, see [1] for details.

Furthermore, we studied agent-based models as continuous-time Markov jump process and their approximations by stochastic differential equations (SDEs). We showed that, under certain conditions, the transfer operator approach bridges the gap between the pathwise results for large populations, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales. The main insights are that the transfer operator approach allows to uncover metastable structures and long timescales associated with rare events for either the considered agent-based model or the SDE processes by means of (many) finite-length trajectories of the corresponding process. For large enough numbers of agents, the metastable structures detected by the transfer operator approach of the agent-based model and SDE process are very close. This implies that the characteristics of the long-term behavior of the agent-based model process can be determined by simulating (many) short trajectories of the SDE process instead, see [2] for details.

Combining the results above, we then showed how the Koopman generator can be used for data-driven model reduction of agent-based dynamics. We demonstrated the techniques using two benchmark agent-based models: an extended voter model acting on different networks and a spatial predator-prey model. We directly learn reduced models from high-dimensional agent-based model data describing the evolution of aggregated states of the system, i.e., modeling the collective behavior of larger groups or the entire population. This method does not require a broad background knowledge of the underlying agent-based model dynamics such that the model can be considered as black box. We showed that the resulting models are good approximations of analytically computed limit SDEs. Furthermore, we analyzed the influence of the connectivity structure on the reduced model and showed that predictions are possible even when no underlying limit process is known. We refer the reader to [3] for more details.

The code for simulating the agent-based model is available at GitHub.

The next goal is to infer reduced surrogate models that can be used for optimization or control.

**Project Webpages**

**Selected Publications
**

- S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte. Data-driven approximation of the Koopman generator: Model reduction, system identification, and control.
*Physica D: Nonlinear Phenomena*, 406:132416, 2020. (Physica D) - J.-H. Niemann, S. Winkelmann, S. Wolf, and C. Schütte. Agent-based modeling: Population limits and large timescales.
*Chaos: An Interdisciplinary Journal of Nonlinear Science*, 31 (3), 2021. - J.-H. Niemann, S. Klus, and C. Schütte. Data-driven model reduction of agent-based systems using the Koopman generator.
*Plos One*, 16 (5), 2020. (Plos One)

**Selected Pictures
**

Figure (a) shows the interaction network of the ABM with 10 agents in three different available types (blue, red yellow). Figure (b) shows a realization of the ABM. Figure (c) shows the relative mean and standard deviation of the theoretical SDE limit model (blue) and the numerically obtained coarse-grained model. Figure (d) shows a heatmap indicating the error for different parameter settings, i.e., number of agents and number of Monte Carlo samples for the estimation with gEDMD.

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