Emergent Data-Driven Model Reductions for Coupled, Heterogeneous Agent-Based Systems

Large collections of coupled, heterogeneous agents can manifest complex dynamical behavior presenting difficulties for simulation and analysis. However, if the collective dynamics lie on a low-dimensional manifold then the original agent-based model may be approximated with a simplified surrogate model on and near the low-dimensional space where the dynamics live. This is typically accomplished by deriving coarse variables that summarize the collective dynamics [1-4], these may take the form of either a collection of scalars or continuous fields (e.g. densities), which are then used as part of a reduced model. Analytically identifying such simplified models is challenging and has traditionally been accomplished through the use of mean-field reductions [5] or an Ott-Antonsen ansatz [6], but is often impossible.

Here we present a data-driven coarse-graining methodology for discovering such emergent, reduced models. We consider two types of reduced models: globally-based models which use global information and predict dynamics using information from the whole ensemble, and locally-based models that use local information, that is, information from just a subset of agents close (close in heterogeneity space, not physical space) to an agent, to predict the dynamics of an agent. For both approaches we are able to learn laws governing the behavior of the reduced system on the low-dimensional manifold directly from time series of states from the agent-based system. These laws take the form of either a system of ordinary differential equations (ODEs), for the globally-based approach, or a partial differential equation (PDE) in the locally-based case. For each technique we employ a specialized artificial neural network integrator that has been templated on an Euler time stepper to learn the laws of the reduced model. As an illustration of the efficacy of our techniques, we consider a simplified all-to-all coupled neuron model for the rhythmic oscillations in the pre-Bötzinger complex and demonstrate how our data-driven surrogate models are able to produce dynamics comparable to the dynamics of the full system. An interesting consequence of our PDE methodology is that our local PDE model is able to learn the behavior of a globally coupled model. This suggests that the neural network learns a coupling topology, different from the actual coupling topology of the agents, that is conducive to modeling the dynamics, although this topology is not directly accessible.

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[2] S. Watanabe and S. H. Strogatz, “Integrability of a globally coupled oscillator array,” Physical review letters 70, 2391 (1993).

[3] K. P. O’Keeffe, H. Hong, and S. H. Strogatz, “Oscillators that sync and swarm,” Nature communications 8, 1–13 (2017).

[4] S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D: Nonlinear Phenomena 143, 1–20 (2000).

[5] C. Bick, M. Goodfellow, C. R. Laing, and E. A. Martens, “Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review,” The Journal of Mathematical Neuroscience 10, 1–43 (2020).

[6] E. Ott and T. M. Antonsen, “Low dimensional behavior of large systems of globally coupled oscillators,” Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 037113 (2008).